Open Case Studies: Predicting Annual Air Pollution

Disclaimer: The purpose of the Open Case Studies project is to demonstrate the use of various data science methods, tools, and software in the context of messy, real-world data. A given case study does not cover all aspects of the research process, is not claiming to be the most appropriate way to analyze a given data set, and should not be used in the context of making policy decisions without external consultation from scientific experts.

This work is licensed under the Creative Commons Attribution-NonCommercial 3.0 (CC BY-NC 3.0) United States License.

Motivation

---

A variety of different sources contribute different types of pollutants to what we call air pollution.

Some sources are natural while others are anthropogenic (human derived):

[source]

Major types of air pollutants

1. Gaseous - Carbon Monoxide (CO), Ozone (O₃), Nitrogen Oxides(NO, NO₂), Sulpher Dioxide (SO₂)
2. Particulate - small liquids and solids suspended in the air (includes lead- can include certain types of dust)
3. Dust - small solids (larger than particulates) that can be suspended in the air for some time but eventually settle
4. Biological - pollen, bacteria, viruses, mold spores

See here for more detail on the types of pollutants in the air.

Particulate pollution

Air pollution particulates are generally described by their size.

There are 3 major categories:

1. Large Coarse Particulate Mater - has diameter of >10 micrometers (10 µm)

2. Coarse Particulate Mater (called PM_10-2.5) - has diameter of between 2.5 µm and 10 µm

3. Fine Particulate Mater (called PM_2.5) - has diameter of < 2.5 µm

PM₁₀ includes any particulate mater <10 µm (both coarse and fine particulate mater)

Here you can see how these sizes compare with a human hair:

[source]

The following plot and table show the relative sizes of these different pollutants in micrometers (µm):

[source]

[source]

This table shows how deeply some of the smaller fine particles can penetrate within the human body:

[source]

Negative impact of particulate exposure on health

Exposure to air pollution is associated with higher rates of mortality in older adults and is known to be a risk factor for many diseases and conditions including but not limited to:

1. Asthma - fine particle exposure (PM_2.5) was found to be associated with higher rates of asthma in children
2. Inflammation in type 1 diabetes - fine particle exposure (PM_2.5) from traffic-related air pollution was associated with increased measures of inflammatory markers in youths with Type 1 diabetes
3. Lung function and emphysema - higher concentrations of ozone (O₃), nitrogen oxides (NO_x), black carbon, and fine particle exposure PM_2.5 , at study baseline were significantly associated with greater increases in percent emphysema per 10 years
4. Low birthweight - fine particle exposure(PM_2.5) was associated with lower birth weight in full-term live births
5. Viral Infection - higher rates of infection and increased severity of infection are associated with higher exposures to pollution levels including fine particle exposure (PM_2.5)

See this review article for more information about sources of air pollution and the influence of air pollution on health.

Sparse monitoring is problematic for Public Health

Historically, epidemiological studies would assess the influence of air pollution on health outcomes by relying on a number of monitors located around the country. However, as can be seen in the following figure, these monitors are relatively sparse in certain regions of the country. Furthermore, dramatic differences in pollution rates can be seen even within the same city.

[source]

This lack of granularity in air pollution monitoring has hindered our ability to discern the full impact of air pollution on health and to identify at-risk locations.

Machine learning offers a solution

An article published in the Environmental Health journal dealt with this issue by using data, including population density, road density, among other features, to model or predict air pollution levels at a more localized scale using machine learning (ML) methods.

Yanosky, J. D. et al. Spatio-temporal modeling of particulate air pollution in the conterminous United States using geographic and meteorological predictors. Environ Health 13, 63 (2014).

The authors of this article state that:

“Exposure to atmospheric particulate matter (PM) remains an important public health concern, although it remains difficult to quantify accurately across large geographic areas with sufficiently high spatial resolution. Recent epidemiologic analyses have demonstrated the importance of spatially- and temporally-resolved exposure estimates, which show larger PM-mediated health effects as compared to nearest monitor or county-specific ambient concentrations.”

The article above demonstrates that machine learning methods can be used to predict air pollution levels when traditional monitoring systems are not available in a particular area or when there is not enough spatial granularity with current monitoring systems. We will use similar methods to predict annual air pollution levels spatially within the US.

Main Question

Our main question:

1. Can we predict annual average air pollution concentrations at the granularity of zip code regional levels using predictors such as data about population density, urbanization, road density, as well as, satellite pollution data and chemical modeling data?

Learning Objectives

In this case study, we will walk you through importing data from CSV files and performing machine learning methods to predict our outcome variable of interest (in this case annual fine particle air pollution estimates).

We will especially focus on using packages and functions from the tidyverse, and more specifically the tidymodels package/ecosystem primarily developed and maintained by Max Kuhn and Davis Vaughan. This package loads more modeling related packages like rsample, recipes, parsnip, yardstick, and dials. We will also briefly cover the workflows and tune packages.

The tidyverse is a library of packages created by RStudio. While some students may be familiar with previous R programming packages, these packages make data science in R especially efficient.

The skills, methods, and concepts that students will be familiar with by the end of this case study are:

Data science skills:

1. Familiarity with the tidymodels ecosystem
2. Ability to evaluate correlation among predictor variables (corrplot and GGally)
3. Ability to implement tidymodels packages such as rsample to split the data into traning and testing sets as well as cross validation sets.
4. Ability to use the recipes, parsnip, and workflows to train and test a linear regression model and random forest model
5. Demonstrate how to visualize geo-spatial data using ggplot2

Statistical concepts and methods:

1. Basic understanding the utility of machine learning for prediction and classification
2. Understanding of the need for training and test set
3. Understanding of the utility of cross validation
4. Understanding of random forest

We will begin by loading the packages that we will need:

library(here)
library(readr)
library(dplyr)
library(skimr)
library(summarytools)
library(magrittr)
library(corrplot)
library(RColorBrewer)
library(GGally)
library(tidymodels)
library(workflows)
library(vip)
library(tune)
library(randomForest)
library(ggplot2)
library(stringr)
library(tidyr)
library(lwgeom) # allows 1263
library(sf)
library(maps)
library(rnaturalearth)
library(rgeos)
library(cowplot)

Package	Use
here	to easily load and save data
readr	to import CSV files
dplyr	to view/arrange/filter/select/compare specific subsets of data
skimr	to get an overview of data
summarytools	to get an overview of data in a different style
magrittr	to use the %<>% pipping operator
corrplot	to make large correlation plots
GGally	to make smaller correlation plots
tidymodels	to load in a set of packages (broom, dials, infer, parsnip, purrr, recipes, rsample, tibble, yardstick)
rsample	to split the data into testing and training sets; to split the training set for cross-validation
recipes	to pre-process data for modeling in a tidy and reproducible way and to extract pre-processed data (major functions are recipe() , prep() and various transformation step_*() functions, as well as juice() - extracts final pre-processed training data and bake() - applies recipe steps to testing data). See here for more info.
parsnip	an interface to create models (major functions are fit(), set_engine())
yardstick	to evaluate the performance of models
broom	to get tidy output for our model fit and performance
ggplot2	to make visualizations with multiple layers
dials	to specify hyper-parameter tuning
tune	to perform cross validation, tune hyper-parameters, and get performance metrics
workflows	to create modeling workflow to streamline the modeling process
vip	to create variable importance plots
randomForest	to perform the random forest analysis
stringr	to manipulate the text the map data
tidyr	to separate data within a column into multiple columns
rnaturalearth	to get the geometry data for the earth to plot the US
maps	to get map database data about counties to draw them on our US map
sf	to convert the map data into a data frame
lwgeom	to use the sf function to convert map geographical data
rgeos	to use geometry data
cowplot	to allow plots to be combined

---

The first time we use a function, we will use the :: to indicate which package we are using. Unless we have overlapping function names, this is not necessary, but we will include it here to be informative about where the functions we will use come from.

Context

The State of Global Air is a report released every year to communicate the impact of air pollution on public health.

The State of Global Air 2019 report which uses data from 2017 stated that:

Air pollution is the fifth leading risk factor for mortality worldwide. It is responsible for more deaths than many better-known risk factors such as malnutrition, alcohol use, and physical inactivity. Each year, more people die from air pollution–related disease than from road traffic injuries or malaria.

[source]

The report also stated that:

In 2017, air pollution is estimated to have contributed to close to 5 million deaths globally — nearly 1 in every 10 deaths.

[source]

The State of Global Air 2018 report using data from 2016 which separated different types of air pollution, found that particulate pollution was particularly associated with mortality.

The 2019 report shows that the highest levels of fine particulate pollution occurs in Africa and Asia and that:

More than 90% of people worldwide live in areas exceeding the World Health Organization (WHO) Guideline for healthy air. More than half live in areas that do not even meet WHO’s least-stringent air quality target.

[source]

Looking at the US specifically, air pollution levels are generally improving, with declining national air pollutant concentration averages as shown from the 2019 Our Nation’s Air report from the US Environmental Protection Agency (EPA):

[source]

However, air pollution continues to contribute to health risk for Americans, in particular in regions with higher than national average rates of pollution that actually at time exceed the WHO’s recommended level. Thus, it is important to obtain high spatial granularity in estimates of air pollution in order to identify locations where populations are experiencing harmful levels of exposure.

You can see that current air quality conditions at this website and you will notice variation across different cities.

For example, here are the conditions in Topeka Kansas at the time this case study was created:

It reports particulate values using what is called the Air Quality Index (AQI). This calculator indicates that 114 AQI is equivalent to 40.7 ug/m³ and is considered unhealthy for sensitive individuals. Thus, some areas exceed the WHO annual exposure guideline (10 ug/m³) and this may adversely affect the health of people living in these locations.

Adverse health effects have been associated with populations experiencing higher pollution exposure despite the levels being below suggested guidelines. Also, it appears that the composition of the particulate mater and the influence of other demographic factors may make specific populations more at risk for adverse health effects due to air pollution. For example, see this article for more details.

The monitor data that we will use in this case study come from a system of monitors in which roughly 90% are located within cities. Hence, there is an equity issue in terms of capturing the air pollution levels of more rural areas. To get a better sense of the pollution exposures for the individuals living in these areas, methods like machine learning can be useful to estimate air pollution levels in areas with little to no monitoring. Specifically, these methods can be used to estimate air pollution in these low monitoring areas so that we can make a map like this where we have annual estimates for all of the contiguous US:

[source]

This is what we aim to achieve in this case study.

Limitations

There are some important considerations regarding the data analysis in this case study to keep in mind:

1. The data do not include information about the composition of particulate mater. Different types of particulates may be more benign or deleterious for health outcomes.

2. Outdoor pollution levels are not necessarily an indication of individual exposures. People spend differing amounts of time indoors and outdoors and are exposed to different pollution levels indoors. Researchers are now developing personal monitoring systems to track air pollution levels on the personal level.

3. Our analysis will use annual mean estimates of pollution levels, but these can vary greatly by season, day and even hour. There are data sources that have finer levels of temporal data, however we are interested in long term exposures, as these appear to be the most influential for health outcomes, so we chose to use annual level data.

What are the data?

---

When using machine learning for prediction, there are two main types of data of interest:

1. An continuous outcome variable that we want to predict
2. A set of feature(s) (or predictor variables) that we use to predict the outcome variable

The outcome variable is what are trying to predict. To build (or train) our model, we use both the outcome and features. The goal is to identify informative features that can explain a large amount of variation in our outcome variable. Using this model, we can then predict the outcome from new observations with the same features where have not observed the outcome.

As a simple example, imagine that we have data about the sales and characteristics of cars from last year and we want to predict which cars might sell well this year. We do not have the sales data yet for this year, but we do know the characteristics of our cars for this year. We can build a model of the characteristics that explained sales last year to estimate what cars might sell well this year. In this case, our outcome variable is the sales of cars, while the different characteristics of the cars make up our features.

Start with a question

This is the most commonly missed step when developing a machine learning algorithm. Machine learning can very easily be turned into an engineering problem. Just dump the outcome and the features into a black box algorithm and viola! But this kind of thinking can lead to major problems. In general good machine learning questions:

1. Have a plausible explanation for why the features predict the outcome.
2. Consider potential variation in both the features and the outcome over time
3. Are consistently re-evaluated on criteria 1 and 2 over time.

In this case study, we want to predict air pollution levels. To build this machine learning algorithm, our outcome variable is fine particulate matter (PM_2.5) captured from air pollution monitors in the contiguous US from 2008. Our features (or predictor variables) include data about population density, road density, urbanization levels, and NASA satellite data. Using this model, we predict air pollution in new areas where there are no monitors.

Our outcome variable

The monitor data that we will be using comes from gravimetric monitors (see picture below) operated by the US Enivornmental Protection Agency (EPA).

These monitors use a filtration system to specifically capture fine particulate matter.

[source]

The weight of this particulate matter is manually measured daily or weekly. For the EPA standard operating procedure for PM gravimetric analysis in 2008, we refer the reader to here.

For more on Gravimetric analysis, you can expand here

Gravimetric analysis is also used for emission testing. The same idea applies: a fresh filter is applied and the desired amount of time passes, then the filter is removed and weighed.

There are other monitoring systems that can provide hourly measurements, but we will not be using data from these monitors in our analysis. Gravimetric analysis is considered to be among the most accurate methods for measuring particulate matter.

In our data set, the value column indicates the PM_2.5 monitor average for 2008 in mass of fine particles/volume of air for 876 gravimetric monitors. The units are micrograms of fine particulate mater (PM) that is less than 2.5 micrometers in diameter per cubic meter of air - mass concentration (ug/m³). Recall the WHO exposure guideline is < 10 ug/m³ on average annually for PM_2.5.

Our features (predictor variables)

There are 48 features with values for each of the 876 monitors (observations). The data comes from the US Enivornmental Protection Agency (EPA), the National Aeronautics and Space Administration (NASA), the US Census, and the National Center for Health Statistics (NCHS).

Click here to see a table about the set of features

Variable	Details
id	Monitor number – the county number is indicated before the decimal – the monitor number is indicated after the decimal Example: 1073.0023 is Jefferson county (1073) and .0023 one of 8 monitors
fips	Federal information processing standard number for the county where the monitor is located – 5 digit id code for counties (zero is often the first value and sometimes is not shown) – the first 2 numbers indicate the state – the last three numbers indicate the county Example: Alabama’s state code is 01 because it is first alphabetically (note: Alaska and Hawaii are not included because they are not part of the contiguous US)
Lat	Latitude of the monitor in degrees
Lon	Longitude of the monitor in degrees
state	State where the monitor is located
county	County where the monitor is located
city	City where the monitor is located
CMAQ	Estimated values of air pollution from a computational model called Community Multiscale Air Quality (CMAQ) – A monitoring system that simulates the physics of the atmosphere using chemistry and weather data to predict the air pollution – Does not use any of the PM2.5 gravimetric monitoring data. (There is a version that does use the gravimetric monitoring data, but not this one!) – Data from the EPA
zcta	Zip Code Tabulation Area where the monitor is located – Postal Zip codes are converted into “generalized areal representations” that are non-overlapping – Data from the 2010 Census
zcta_area	Land area of the zip code area in meters squared – Data from the 2010 Census
zcta_pop	Population in the zip code area – Data from the 2010 Census
imp_a500	Impervious surface measure – Within a circle with a radius of 500 meters around the monitor – Impervious surface are roads, concrete, parking lots, buildings – This is a measure of development
imp_a1000	Impervious surface measure – Within a circle with a radius of 1000 meters around the monitor
imp_a5000	Impervious surface measure – Within a circle with a radius of 5000 meters around the monitor
imp_a10000	Impervious surface measure – Within a circle with a radius of 10000 meters around the monitor
imp_a15000	Impervious surface measure – Within a circle with a radius of 15000 meters around the monitor
county_area	Land area of the county of the monitor in meters squared
county_pop	Population of the county of the monitor
Log_dist_to_prisec	Log (Natural log) distance to a primary or secondary road from the monitor – Highway or major road
log_pri_length_5000	Count of primary road length in meters in a circle with a radius of 5000 meters around the monitor (Natural log) – Highways only
log_pri_length_10000	Count of primary road length in meters in a circle with a radius of 10000 meters around the monitor (Natural log) – Highways only
log_pri_length_15000	Count of primary road length in meters in a circle with a radius of 15000 meters around the monitor (Natural log) – Highways only
log_pri_length_25000	Count of primary road length in meters in a circle with a radius of 25000 meters around the monitor (Natural log) – Highways only
log_prisec_length_500	Count of primary and secondary road length in meters in a circle with a radius of 500 meters around the monitor (Natural log) – Highway and secondary roads
log_prisec_length_1000	Count of primary and secondary road length in meters in a circle with a radius of 1000 meters around the monitor (Natural log) – Highway and secondary roads
log_prisec_length_5000	Count of primary and secondary road length in meters in a circle with a radius of 5000 meters around the monitor (Natural log) – Highway and secondary roads
log_prisec_length_10000	Count of primary and secondary road length in meters in a circle with a radius of 10000 meters around the monitor (Natural log) – Highway and secondary roads
log_prisec_length_15000	Count of primary and secondary road length in meters in a circle with a radius of 15000 meters around the monitor (Natural log) – Highway and secondary roads
log_prisec_length_25000	Count of primary and secondary road length in meters in a circle with a radius of 25000 meters around the monitor (Natural log) – Highway and secondary roads
log_nei_2008_pm25_sum_10000	Tons of emissions from major sources data base (annual data) sum of all sources within a circle with a radius of 10000 meters of distance around the monitor (Natural log)
log_nei_2008_pm25_sum_15000	Tons of emissions from major sources data base (annual data) sum of all sources within a circle with a radius of 15000 meters of distance around the monitor (Natural log)
log_nei_2008_pm25_sum_25000	Tons of emissions from major sources data base (annual data) sum of all sources within a circle with a radius of 25000 meters of distance around the monitor (Natural log)
log_nei_2008_pm10_sum_10000	Tons of emissions from major sources data base (annual data) sum of all sources within a circle with a radius of 10000 meters of distance around the monitor (Natural log)
log_nei_2008_pm10_sum_15000	Tons of emissions from major sources data base (annual data) sum of all sources within a circle with a radius of 15000 meters of distance around the monitor (Natural log)
log_nei_2008_pm10_sum_25000	Tons of emissions from major sources data base (annual data) sum of all sources within a circle with a radius of 25000 meters of distance around the monitor (Natural log)
popdens_county	Population density (number of people per kilometer squared area of the county)
popdens_zcta	Population density (number of people per kilometer squared area of zcta)
nohs	Percentage of people in zcta area where the monitor is that do not have a high school degree – Data from the Census
somehs	Percentage of people in zcta area where the monitor whose highest formal educational attainment was some high school education – Data from the Census
hs	Percentage of people in zcta area where the monitor whose highest formal educational attainment was completing a high school degree – Data from the Census
somecollege	Percentage of people in zcta area where the monitor whose highest formal educational attainment was completing some college education – Data from the Census
associate	Percentage of people in zcta area where the monitor whose highest formal educational attainment was completing an associate degree – Data from the Census
bachelor	Percentage of people in zcta area where the monitor whose highest formal educational attainment was a bachelor’s degree – Data from the Census
grad	Percentage of people in zcta area where the monitor whose highest formal educational attainment was a graduate degree – Data from the Census
pov	Percentage of people in zcta area where the monitor is that lived in poverty in 2008 - or would it have been 2007 guidelines??https://aspe.hhs.gov/2007-hhs-poverty-guidelines – Data from the Census
hs_orless	Percentage of people in zcta area where the monitor whose highest formal educational attainment was a high school degree or less (sum of nohs, somehs, and hs)
urc2013	2013 Urban-rural classification of the county where the monitor is located – 6 category variable - 1 is totally urban 6 is completely rural – Data from the National Center for Health Statistics](https://www.cdc.gov/nchs/index.htm){target="_blank"}
urc2006	2006 Urban-rural classification of the county where the monitor is located – 6 category variable - 1 is totally urban 6 is completely rural – Data from the National Center for Health Statistics
aod	Aerosol Optical Depth measurement from a NASA satellite – based on the diffraction of a laser – used as a proxy of particulate pollution – unit-less - higher value indicates more pollution – Data from NASA

Many of these features have to do with the circular area around the monitor called the “buffer”. These are illustrated in the following figure:

[source]

Data Import

---

We have one CSV file that contains both our single outcome variable and all of our features (or predictor variables).

Next, we import our data into R now so that we can explore the data further. We will call our data object pm for particulate matter. We import the data using the read_csv() function from the readr package.

pm <- readr::read_csv(here("docs", "pm25_data.csv"))

Data Exploration and Wrangling

---

The first step in performing any data analysis is to explore the data. For example, we might want to better understand the variables included in the data, as we may learn about important details about the data that we should keep in mind as we try to predict our outcome variable.

First, let’s just get a general sense of our data. We can do that using the glimpse() function of the dplyr package (it is also in the tibble package).

We will also use the %>% pipe, which can be used to define the input for later sequential steps. This will make more sense when we have multiple sequential steps using the same data object. To use the pipe notation we need to install and load dplyr as well.

For example, here we start with pm data object and “pipe” the object into as input into the glimpse() function. The output is is an overview of what is in the pm object such as the number of rows and columns, all the column names, the data types for each column and the first view values in each column. The output below is scrollable so you can see everything from the glimpse() function.

# Scroll through the output!
pm %>%
  dplyr::glimpse()

Rows: 876
Columns: 50
$ id                          <dbl> 1003.001, 1027.000, 1033.100, 1049.100, 1…
$ value                       <dbl> 9.597647, 10.800000, 11.212174, 11.659091…
$ fips                        <dbl> 1003, 1027, 1033, 1049, 1055, 1069, 1073,…
$ lat                         <dbl> 30.49800, 33.28126, 34.75878, 34.28763, 3…
$ lon                         <dbl> -87.88141, -85.80218, -87.65056, -85.9683…
$ state                       <chr> "Alabama", "Alabama", "Alabama", "Alabama…
$ county                      <chr> "Baldwin", "Clay", "Colbert", "DeKalb", "…
$ city                        <chr> "Fairhope", "Ashland", "Muscle Shoals", "…
$ CMAQ                        <dbl> 8.098836, 9.766208, 9.402679, 8.534772, 9…
$ zcta                        <dbl> 36532, 36251, 35660, 35962, 35901, 36303,…
$ zcta_area                   <dbl> 190980522, 374132430, 16716984, 203836235…
$ zcta_pop                    <dbl> 27829, 5103, 9042, 8300, 20045, 30217, 90…
$ imp_a500                    <dbl> 0.01730104, 1.96972318, 19.17301038, 5.78…
$ imp_a1000                   <dbl> 1.4096021, 0.8531574, 11.1448962, 3.86764…
$ imp_a5000                   <dbl> 3.3360118, 0.9851479, 15.1786154, 1.23114…
$ imp_a10000                  <dbl> 1.9879187, 0.5208189, 9.7253870, 1.031646…
$ imp_a15000                  <dbl> 1.4386207, 0.3359198, 5.2472094, 0.973044…
$ county_area                 <dbl> 4117521611, 1564252280, 1534877333, 20126…
$ county_pop                  <dbl> 182265, 13932, 54428, 71109, 104430, 1015…
$ log_dist_to_prisec          <dbl> 4.648181, 7.219907, 5.760131, 3.721489, 5…
$ log_pri_length_5000         <dbl> 8.517193, 8.517193, 8.517193, 8.517193, 9…
$ log_pri_length_10000        <dbl> 9.210340, 9.210340, 9.274303, 10.409411, …
$ log_pri_length_15000        <dbl> 9.630228, 9.615805, 9.658899, 11.173626, …
$ log_pri_length_25000        <dbl> 11.32735, 10.12663, 10.15769, 11.90959, 1…
$ log_prisec_length_500       <dbl> 7.295356, 6.214608, 8.611945, 7.310155, 8…
$ log_prisec_length_1000      <dbl> 8.195119, 7.600902, 9.735569, 8.585843, 9…
$ log_prisec_length_5000      <dbl> 10.815042, 10.170878, 11.770407, 10.21420…
$ log_prisec_length_10000     <dbl> 11.88680, 11.40554, 12.84066, 11.50894, 1…
$ log_prisec_length_15000     <dbl> 12.205723, 12.042963, 13.282656, 12.35366…
$ log_prisec_length_25000     <dbl> 13.41395, 12.79980, 13.79973, 13.55979, 1…
$ log_nei_2008_pm25_sum_10000 <dbl> 0.318035438, 3.218632928, 6.573127301, 0.…
$ log_nei_2008_pm25_sum_15000 <dbl> 1.967358961, 3.218632928, 6.581917457, 3.…
$ log_nei_2008_pm25_sum_25000 <dbl> 5.067308, 3.218633, 6.875900, 4.887665, 4…
$ log_nei_2008_pm10_sum_10000 <dbl> 1.35588511, 3.31111648, 6.69187313, 0.000…
$ log_nei_2008_pm10_sum_15000 <dbl> 2.26783411, 3.31111648, 6.70127741, 3.350…
$ log_nei_2008_pm10_sum_25000 <dbl> 5.628728, 3.311116, 7.148858, 5.171920, 4…
$ popdens_county              <dbl> 44.265706, 8.906492, 35.460814, 35.330814…
$ popdens_zcta                <dbl> 145.716431, 13.639555, 540.887040, 40.718…
$ nohs                        <dbl> 3.3, 11.6, 7.3, 14.3, 4.3, 5.8, 7.1, 2.7,…
$ somehs                      <dbl> 4.9, 19.1, 15.8, 16.7, 13.3, 11.6, 17.1, …
$ hs                          <dbl> 25.1, 33.9, 30.6, 35.0, 27.8, 29.8, 37.2,…
$ somecollege                 <dbl> 19.7, 18.8, 20.9, 14.9, 29.2, 21.4, 23.5,…
$ associate                   <dbl> 8.2, 8.0, 7.6, 5.5, 10.1, 7.9, 7.3, 8.0, …
$ bachelor                    <dbl> 25.3, 5.5, 12.7, 7.9, 10.0, 13.7, 5.9, 17…
$ grad                        <dbl> 13.5, 3.1, 5.1, 5.8, 5.4, 9.8, 2.0, 8.7, …
$ pov                         <dbl> 6.1, 19.5, 19.0, 13.8, 8.8, 15.6, 25.5, 7…
$ hs_orless                   <dbl> 33.3, 64.6, 53.7, 66.0, 45.4, 47.2, 61.4,…
$ urc2013                     <dbl> 4, 6, 4, 6, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2,…
$ urc2006                     <dbl> 5, 6, 4, 5, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2,…
$ aod                         <dbl> 37.36364, 34.81818, 36.00000, 33.08333, 4…

We can see that there are 876 monitors (rows) and that we have 50 total variables (columns) - one of which is the outcome variable. In this case, the outcome variable is called value.

AVOCADO: Should we link to a code book here? Like could we use this as a teaching opportunity to demonstrate the purpose of a code book so a student knows how we know the PM values are value? avocado comment: there isnt a code book related to this data, Roger wrangled it for me, but the vaping case study actually uses code books for that data. Would you like to link to a generic code book and describe that process?

Notice that some of the variables that we would think of as factors (or categorical data) are currently of class character as indicated by the <chr> just to the right of the column names/variable names in the glimpse() output. This means that the variable values are character strings, such as words or phrases.

The other variables are of class <dbl>, which stands for double precision which indicates that the are numeric and that they have decimal values. In contrast, one could have integer values which would not allow for decimal numbers. Here is a link for more information on double precision numeric values.

Another common data class is factor which is abreviated like this: <fct>. A factor is something that has unique levels but there is no appreciable order to the levels. For example we can have a numeric value that is just an id that we want to be interpreted as just a unique level and not as the number that it would typically indicate. This would be useful for several of our variables:

1. the monitor ID (id)
2. the Federal Information Processing Standard number for the county where the monitor was located (fips)
3. the zip code tabulation area (zcta)

None of the values actually have any real numeric meaning, so we want to make sure that R does not interpret them as if they do.

So let’s convert these variables into factors. We can do this using the across() function of the dplyr package and the as.factor() base function. The across() function has two main arguments: (i) the columns you want to operate on and (ii) the function or list of functions to apply to each column.

In this case, we are also using the magrittr assignment pipe or double pipe that looks like this %<>% of the magrittr package. This allows us use the pm data as input, but also reassigns the output to the same data object name.

# Scroll through the output!
pm %<>%
  mutate(across(c(id, fips, zcta), as.factor)) 

glimpse(pm)

Rows: 876
Columns: 50
$ id                          <fct> 1003.001, 1027.0001, 1033.1002, 1049.1003…
$ value                       <dbl> 9.597647, 10.800000, 11.212174, 11.659091…
$ fips                        <fct> 1003, 1027, 1033, 1049, 1055, 1069, 1073,…
$ lat                         <dbl> 30.49800, 33.28126, 34.75878, 34.28763, 3…
$ lon                         <dbl> -87.88141, -85.80218, -87.65056, -85.9683…
$ state                       <chr> "Alabama", "Alabama", "Alabama", "Alabama…
$ county                      <chr> "Baldwin", "Clay", "Colbert", "DeKalb", "…
$ city                        <chr> "Fairhope", "Ashland", "Muscle Shoals", "…
$ CMAQ                        <dbl> 8.098836, 9.766208, 9.402679, 8.534772, 9…
$ zcta                        <fct> 36532, 36251, 35660, 35962, 35901, 36303,…
$ zcta_area                   <dbl> 190980522, 374132430, 16716984, 203836235…
$ zcta_pop                    <dbl> 27829, 5103, 9042, 8300, 20045, 30217, 90…
$ imp_a500                    <dbl> 0.01730104, 1.96972318, 19.17301038, 5.78…
$ imp_a1000                   <dbl> 1.4096021, 0.8531574, 11.1448962, 3.86764…
$ imp_a5000                   <dbl> 3.3360118, 0.9851479, 15.1786154, 1.23114…
$ imp_a10000                  <dbl> 1.9879187, 0.5208189, 9.7253870, 1.031646…
$ imp_a15000                  <dbl> 1.4386207, 0.3359198, 5.2472094, 0.973044…
$ county_area                 <dbl> 4117521611, 1564252280, 1534877333, 20126…
$ county_pop                  <dbl> 182265, 13932, 54428, 71109, 104430, 1015…
$ log_dist_to_prisec          <dbl> 4.648181, 7.219907, 5.760131, 3.721489, 5…
$ log_pri_length_5000         <dbl> 8.517193, 8.517193, 8.517193, 8.517193, 9…
$ log_pri_length_10000        <dbl> 9.210340, 9.210340, 9.274303, 10.409411, …
$ log_pri_length_15000        <dbl> 9.630228, 9.615805, 9.658899, 11.173626, …
$ log_pri_length_25000        <dbl> 11.32735, 10.12663, 10.15769, 11.90959, 1…
$ log_prisec_length_500       <dbl> 7.295356, 6.214608, 8.611945, 7.310155, 8…
$ log_prisec_length_1000      <dbl> 8.195119, 7.600902, 9.735569, 8.585843, 9…
$ log_prisec_length_5000      <dbl> 10.815042, 10.170878, 11.770407, 10.21420…
$ log_prisec_length_10000     <dbl> 11.88680, 11.40554, 12.84066, 11.50894, 1…
$ log_prisec_length_15000     <dbl> 12.205723, 12.042963, 13.282656, 12.35366…
$ log_prisec_length_25000     <dbl> 13.41395, 12.79980, 13.79973, 13.55979, 1…
$ log_nei_2008_pm25_sum_10000 <dbl> 0.318035438, 3.218632928, 6.573127301, 0.…
$ log_nei_2008_pm25_sum_15000 <dbl> 1.967358961, 3.218632928, 6.581917457, 3.…
$ log_nei_2008_pm25_sum_25000 <dbl> 5.067308, 3.218633, 6.875900, 4.887665, 4…
$ log_nei_2008_pm10_sum_10000 <dbl> 1.35588511, 3.31111648, 6.69187313, 0.000…
$ log_nei_2008_pm10_sum_15000 <dbl> 2.26783411, 3.31111648, 6.70127741, 3.350…
$ log_nei_2008_pm10_sum_25000 <dbl> 5.628728, 3.311116, 7.148858, 5.171920, 4…
$ popdens_county              <dbl> 44.265706, 8.906492, 35.460814, 35.330814…
$ popdens_zcta                <dbl> 145.716431, 13.639555, 540.887040, 40.718…
$ nohs                        <dbl> 3.3, 11.6, 7.3, 14.3, 4.3, 5.8, 7.1, 2.7,…
$ somehs                      <dbl> 4.9, 19.1, 15.8, 16.7, 13.3, 11.6, 17.1, …
$ hs                          <dbl> 25.1, 33.9, 30.6, 35.0, 27.8, 29.8, 37.2,…
$ somecollege                 <dbl> 19.7, 18.8, 20.9, 14.9, 29.2, 21.4, 23.5,…
$ associate                   <dbl> 8.2, 8.0, 7.6, 5.5, 10.1, 7.9, 7.3, 8.0, …
$ bachelor                    <dbl> 25.3, 5.5, 12.7, 7.9, 10.0, 13.7, 5.9, 17…
$ grad                        <dbl> 13.5, 3.1, 5.1, 5.8, 5.4, 9.8, 2.0, 8.7, …
$ pov                         <dbl> 6.1, 19.5, 19.0, 13.8, 8.8, 15.6, 25.5, 7…
$ hs_orless                   <dbl> 33.3, 64.6, 53.7, 66.0, 45.4, 47.2, 61.4,…
$ urc2013                     <dbl> 4, 6, 4, 6, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2,…
$ urc2006                     <dbl> 5, 6, 4, 5, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2,…
$ aod                         <dbl> 37.36364, 34.81818, 36.00000, 33.08333, 4…

Great! Now we can see that these variables are now factors as indicated by <fct> after the variable name.

Packages to get a sense of the data

The skim() function of the skimr package is also really helpful for getting a general sense of your data. By design, it provies summary statistics about variables in the data set. designed to provide summary statistics about variables.

# Scroll through the output!
skim(pm)

Data summary

Name	pm
Number of rows	876
Number of columns	50
_______________________
Column type frequency:
character	3
factor	3
numeric	44
________________________
Group variables	None

Variable type: character

skim_variable	n_missing	complete_rate	min	max	empty	n_unique	whitespace
state	0	1	4	20	0	49	0
county	0	1	3	20	0	471	0
city	0	1	4	48	0	607	0

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
id	0	1	FALSE	876	100: 1, 102: 1, 103: 1, 104: 1
fips	0	1	FALSE	569	170: 12, 603: 10, 261: 9, 107: 8
zcta	0	1	FALSE	842	475: 3, 110: 2, 160: 2, 290: 2

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
value	0	1	10.81	2.580000e+00	3.02	9.27	11.15	12.37	2.316000e+01	▂▆▇▁▁
lat	0	1	38.48	4.620000e+00	25.47	35.03	39.30	41.66	4.840000e+01	▁▃▅▇▂
lon	0	1	-91.74	1.496000e+01	-124.18	-99.16	-87.47	-80.69	-6.804000e+01	▃▂▃▇▃
CMAQ	0	1	8.41	2.970000e+00	1.63	6.53	8.62	10.24	2.313000e+01	▃▇▃▁▁
zcta_area	0	1	183173481.91	5.425989e+08	15459.00	14204601.75	37653560.50	160041508.25	8.164821e+09	▇▁▁▁▁
zcta_pop	0	1	24227.58	1.777216e+04	0.00	9797.00	22014.00	35004.75	9.539700e+04	▇▇▃▁▁
imp_a500	0	1	24.72	1.934000e+01	0.00	3.70	25.12	40.22	6.961000e+01	▇▅▆▃▂
imp_a1000	0	1	24.26	1.802000e+01	0.00	5.32	24.53	38.59	6.750000e+01	▇▅▆▃▁
imp_a5000	0	1	19.93	1.472000e+01	0.05	6.79	19.07	30.11	7.460000e+01	▇▆▃▁▁
imp_a10000	0	1	15.82	1.381000e+01	0.09	4.54	12.36	24.17	7.209000e+01	▇▃▂▁▁
imp_a15000	0	1	13.43	1.312000e+01	0.11	3.24	9.67	20.55	7.110000e+01	▇▃▁▁▁
county_area	0	1	3768701992.12	6.212830e+09	33703512.00	1116536297.50	1690826566.50	2878192209.00	5.194723e+10	▇▁▁▁▁
county_pop	0	1	687298.44	1.293489e+06	783.00	100948.00	280730.50	743159.00	9.818605e+06	▇▁▁▁▁
log_dist_to_prisec	0	1	6.19	1.410000e+00	-1.46	5.43	6.36	7.15	1.045000e+01	▁▁▃▇▁
log_pri_length_5000	0	1	9.82	1.080000e+00	8.52	8.52	10.05	10.73	1.205000e+01	▇▂▆▅▂
log_pri_length_10000	0	1	10.92	1.130000e+00	9.21	9.80	11.17	11.83	1.302000e+01	▇▂▇▇▃
log_pri_length_15000	0	1	11.50	1.150000e+00	9.62	10.87	11.72	12.40	1.359000e+01	▆▂▇▇▃
log_pri_length_25000	0	1	12.24	1.100000e+00	10.13	11.69	12.46	13.05	1.436000e+01	▅▃▇▇▃
log_prisec_length_500	0	1	6.99	9.500000e-01	6.21	6.21	6.21	7.82	9.400000e+00	▇▁▂▂▁
log_prisec_length_1000	0	1	8.56	7.900000e-01	7.60	7.60	8.66	9.20	1.047000e+01	▇▅▆▃▁
log_prisec_length_5000	0	1	11.28	7.800000e-01	8.52	10.91	11.42	11.83	1.278000e+01	▁▁▃▇▃
log_prisec_length_10000	0	1	12.41	7.300000e-01	9.21	11.99	12.53	12.94	1.385000e+01	▁▁▃▇▅
log_prisec_length_15000	0	1	13.03	7.200000e-01	9.62	12.59	13.13	13.57	1.441000e+01	▁▁▃▇▅
log_prisec_length_25000	0	1	13.82	7.000000e-01	10.13	13.38	13.92	14.35	1.523000e+01	▁▁▃▇▆
log_nei_2008_pm25_sum_10000	0	1	3.97	2.350000e+00	0.00	2.15	4.29	5.69	9.120000e+00	▆▅▇▆▂
log_nei_2008_pm25_sum_15000	0	1	4.72	2.250000e+00	0.00	3.47	5.00	6.35	9.420000e+00	▃▃▇▇▂
log_nei_2008_pm25_sum_25000	0	1	5.67	2.110000e+00	0.00	4.66	5.91	7.28	9.650000e+00	▂▂▇▇▃
log_nei_2008_pm10_sum_10000	0	1	4.35	2.320000e+00	0.00	2.69	4.62	6.07	9.340000e+00	▅▅▇▇▂
log_nei_2008_pm10_sum_15000	0	1	5.10	2.180000e+00	0.00	3.87	5.39	6.72	9.710000e+00	▂▃▇▇▂
log_nei_2008_pm10_sum_25000	0	1	6.07	2.010000e+00	0.00	5.10	6.37	7.52	9.880000e+00	▁▂▆▇▃
popdens_county	0	1	551.76	1.711510e+03	0.26	40.77	156.67	510.81	2.682191e+04	▇▁▁▁▁
popdens_zcta	0	1	1279.66	2.757490e+03	0.00	101.15	610.35	1382.52	3.041884e+04	▇▁▁▁▁
nohs	0	1	6.99	7.210000e+00	0.00	2.70	5.10	8.80	1.000000e+02	▇▁▁▁▁
somehs	0	1	10.17	6.200000e+00	0.00	5.90	9.40	13.90	7.220000e+01	▇▂▁▁▁
hs	0	1	30.32	1.140000e+01	0.00	23.80	30.75	36.10	1.000000e+02	▂▇▂▁▁
somecollege	0	1	21.58	8.600000e+00	0.00	17.50	21.30	24.70	1.000000e+02	▆▇▁▁▁
associate	0	1	7.13	4.010000e+00	0.00	4.90	7.10	8.80	7.140000e+01	▇▁▁▁▁
bachelor	0	1	14.90	9.710000e+00	0.00	8.80	12.95	19.22	1.000000e+02	▇▂▁▁▁
grad	0	1	8.91	8.650000e+00	0.00	3.90	6.70	11.00	1.000000e+02	▇▁▁▁▁
pov	0	1	14.95	1.133000e+01	0.00	6.50	12.10	21.22	6.590000e+01	▇▅▂▁▁
hs_orless	0	1	47.48	1.675000e+01	0.00	37.92	48.65	59.10	1.000000e+02	▁▃▇▃▁
urc2013	0	1	2.92	1.520000e+00	1.00	2.00	3.00	4.00	6.000000e+00	▇▅▃▂▁
urc2006	0	1	2.97	1.520000e+00	1.00	2.00	3.00	4.00	6.000000e+00	▇▅▃▂▁
aod	0	1	43.70	1.956000e+01	5.00	31.66	40.17	49.67	1.430000e+02	▃▇▁▁▁

Notice how there is a column called n_missing about the number of values that are missing. This is also indicated by the complete_rate variable (or missing/number of observations). In our data set, it looks like our data do not contain any missing data.

Also notice how the function provides separate tables of summary statistics for each data type: character, factor and numeric.

Next, the n_unqiue column shows us the number of unique values for each of our columns. We can see that there are 49 states represented in the data.

We can see that for many variables there are many low values as the distribution shows two peaks, one near zero and another with a higher value. This is true for the imp variables (measures of development), the nei variables (measures of emission sources) and the road density variables. We can also see that the range of some of the variables is very large, in particular the area and population related variables.

Let’s take a look to see which states are included:

pm %>% 
  distinct(state) 

Scroll through the output:

# A tibble: 49 x 1
   state               
   <chr>               
 1 Alabama             
 2 Arizona             
 3 Arkansas            
 4 California          
 5 Colorado            
 6 Connecticut         
 7 Delaware            
 8 District Of Columbia
 9 Florida             
10 Georgia             
11 Idaho               
12 Illinois            
13 Indiana             
14 Iowa                
15 Kansas              
16 Kentucky            
17 Louisiana           
18 Maine               
19 Maryland            
20 Massachusetts       
21 Michigan            
22 Minnesota           
23 Mississippi         
24 Missouri            
25 Montana             
26 Nebraska            
27 Nevada              
28 New Hampshire       
29 New Jersey          
30 New Mexico          
31 New York            
32 North Carolina      
33 North Dakota        
34 Ohio                
35 Oklahoma            
36 Oregon              
37 Pennsylvania        
38 Rhode Island        
39 South Carolina      
40 South Dakota        
41 Tennessee           
42 Texas               
43 Utah                
44 Vermont             
45 Virginia            
46 Washington          
47 West Virginia       
48 Wisconsin           
49 Wyoming             

Ah, it looks like “District of Columbia” is being included as a state. We can see that indeed Alaska and Hawaii are not included in the data.

Evaluate correlation among possible predictors

In prediction analyses, it is also useful to evaluate if any of the variables are correlated. Why should we care about this?

If we are using a regression to model our data then we might run into a problem called multicolinearity which can lead us to misinterpret what is really predictive of our response variable. This phenomenon occurs when the predictor variables can actually predict one another. See this case study for a deeper explanation about this. Another reason we should look out for correlation is that we dont want to include redudant variables. This can add unnecessary noise to our algorithm causing a reduction in prediction accuracy and it can cause our algroithm to be unnecessarily slower. Finally, it can also make it difficult to interpret what variables are actually predictive.

Intuitively we can expect some of our variables to be correlated.

Let’s first take a look at all of our numeric variabels with thecorrplot package: The corrplot package is another option to look at correlation among possible predictors, and particularly useful if we have many predictors.

First, we calculate the Pearson correlation coefficients between all features pairwise using the cor() function of the stats package (which is loaded automatically). Then we use the corrplot::corrplot() function.

PM_cor <- cor(pm %>% dplyr::select_if(is.numeric))
corrplot::corrplot(PM_cor, tl.cex = 0.5)

The tl.cex = 0.5 argument controls the size of the text label.

We can also plot the absolute value of the Pearson correlation coefficients using the abs() function from base R and change the order of the columns.

corrplot(abs(PM_cor), order = "hclust", tl.cex = 0.5, cl.lim = c(0, 1))

There are several options for ordering the variables. See here for more options. Here we will use the “hclust” option for ordering by hierarchical clustering - which will order the variables by how similar they are to one another.

The cl.lim = c(0, 1) argument limits the color label to be between 0 and 1.

We can see that the development variables (imp) variables are correlated with each other as we might expect. We also see that the road density variables seem to be correlated with each other, and the emission variables seem to be correlated with each other.

Also notice that none of the predictors are highly correlated with our outcome variable (value).

We can take also take a closer look using the ggcorr() function and the ggpairs() function of the GGally package.

To select our variables of interest we can use the select() function with the contains() function of the tidyr package.

First let’s look at the imp/development variables. We can change the default color palette (palette = "RdBu") and add on correlation coefficients to the plot (label = TRUE).

select(pm, contains("imp")) %>%
  ggcorr(palette = "RdBu", label = TRUE)

select(pm, contains("imp")) %>%
  ggpairs()

Indeed, we can see that imp_a1000 and imp_a500 are highly correlated, as well as imp_a10000, imp_a15000.

Next, let’s take a look at the road density data:

select(pm, contains("pri")) %>%
  ggcorr(palette = "RdBu", hjust = .85, size = 3,
       layout.exp=2, label = TRUE)

We can see that many of the road density variables are highly correlated with one another, while others are less so.

Finally let’s look at the emission variables.

select(pm, contains("nei")) %>%
  ggcorr(palette = "RdBu", hjust = .85, size = 3,
       layout.exp=2, label = TRUE)

select(pm, contains("nei")) %>%
  ggpairs()

We would also expect the population density data might correlate with some of these variables. Let’s take a look.

pm %>%
select(log_nei_2008_pm25_sum_10000, popdens_county, 
       log_pri_length_10000, imp_a10000) %>%
  ggcorr(palette = "RdBu",  hjust = .85, size = 3,
       layout.exp=2, label = TRUE)

pm %>%
select(log_nei_2008_pm25_sum_10000, popdens_county, 
       log_pri_length_10000, imp_a10000, county_pop) %>%
  ggpairs()

Interesting, so these variables don’t appear to be highly correlated, therefore we might need variables from each of the categories to predict our monitor PM_2.5 pollution values.

Because some variables in our data have extreme values, it might be good to take a log transformation. This can affect our estimates of correlation.

pm %>%
  mutate(log_popdens_county= log(popdens_county)) %>%
select(log_nei_2008_pm25_sum_10000, log_popdens_county, 
       log_pri_length_10000, imp_a10000) %>%
  ggcorr(palette = "RdBu",  hjust = .85, size = 3,
       layout.exp=2, label = TRUE)

pm %>%
  mutate(log_popdens_county= log(popdens_county)) %>%
  mutate(log_pop_county = log(county_pop)) %>%
select(log_nei_2008_pm25_sum_10000, log_popdens_county, 
       log_pri_length_10000, imp_a10000, log_pop_county) %>%
  ggpairs()

Indeed this increased the correlation, but variables from each of these categories may still prove to be useful for prediction.

Now that we have a sense of what our data are, we can get started with building a machine learning model to predict air pollution.

What is machine learning?

---

You may have learned about the central dogma of statistics that you sample from a population

Then you try to guess what will happen in the population from the sample.

For prediction we have a similar sampling problem

But now we are trying to build a rule that can be used to predict a single observation’s value of some characteristic using the others.

Let’s make this more concrete.

If you recall from the What are the data? section above, when we are using machine learning for prediction, our data consists of:

1. An continuous outcome variable that we want to predict
2. A set of feature(s) (or predictor variables) that we use to predict the outcome variable

We will use \(Y\) to denote the outcome variable and \(X = (X_1, \dots, X_p)\) to denote \(p\) different features (or predictor variables). Because our outcome variable is continuous (as opposed to categorical), we are interested in a particular type of machine learning algorithm.

Our goal is to build a machine learning algorithm uses the features \(X\) as input and predicts a outcome variable (or air pollution levels) in the situation where we do not know the outcome variable.

The way we do this is to use data where we have both the features \((X_1=x_1, \dots X_p=x_p)\) and the actual outcome \(Y\) data to train a machine learning algorithm to predict the outcome, which we call \(\hat{Y}\). When we say train a machine learning algorithm we mean estimate a function \(f\) that uses the predictor variables \(X\) as input or \(\hat{Y} = f(X)\).

ML as an optimization problem

If we are doing a good job, then our predicted outcome \(\hat{Y}\) should closely match our actual outcome \(Y\) that we observed.

In this way, we can think of machine learning (ML) as an optimization problem that tries to minimize the distance between \(\hat{Y} = f(X)\) and \(Y\).

\[d(Y - f(X))\] The choice of distance metric \(d(\cdot)\) can be the mean of the absolute or squared difference or something more complicated. Much of the fields of statistics and computer science are focused on defining \(f\) and \(d\).

The parts of an ML problem

To set up a machine learning (ML) problem, we need a few components. To solve a (standard) machine learning problem you need:

1. A data set to train from.
2. An algorithm or set of algorithms you can use to try values of \(f\)
3. A distance metric \(d\) for measuring how close \(Y\) is to \(\hat{Y}\)
4. A definition of what a “good” distance is

While each of these components is a technical problem, there has been a ton of work addressing those technical details. The most pressing open issue in machine learning is realizing that though these are technical steps they are not objective steps. In other words, how you choose the data, algorithm, metric, and definition of “good” says what you value and can dramatically change the results. A couple of cases where this was a big deal are:

1. Machine learning for recidivism - people built ML models to predict who would re-commit a crime. But these predictions were based on historically biased data which led to biased predictions about who would commit new crimes.
2. Deciding how self driving cars should act - self driving cars will have to make decisions about how to drive, who they might injure, and how to avoid accidents. Depending on our choices for \(f\) and \(d\) these might lead to wildly different kinds of self driving cars. Try out the moralmachine to see how this looks in practice.

Now that we know a bit more about machine learning, let’s build a model to predict air pollution levels using the tidymodels framework.

Machine learning with tidymodels

---

The goal is to build a machine learning algorithm uses the features as input and predicts a outcome variable (or air pollution levels) in the situation where we do not know the outcome variable. The way we do this is to use data where we have both the input and output data to train a machine learning algorithm. To train a machine learning algorithm, we will use the tidymodels package ecosystem.

Overview

The tidymodels ecosystem

To perform our analysis we will be using the tidymodels suite of packages. You may be familiar with the older packages caret or mlr which are also for machine learning and modeling but are not a part of the tidyverse. Max Kuhn describes tidymodels like this:

“Other packages, such as caret and mlr, help to solve the R model API issue. These packages do a lot of other things too: pre-processing, model tuning, resampling, feature selection, ensembling, and so on. In the tidyverse, we strive to make our packages modular and parsnip is designed only to solve the interface issue. It is not designed to be a drop-in replacement for caret. The tidymodels package collection, which includes parsnip, has other packages for many of these tasks, and they are designed to work together. We are working towards higher-level APIs that can replicate and extend what the current model packages can do.”

There are many R packages in the tidymodels ecosystem, which assist with various steps in the process of building a machine learning algorithm:

AVOCADO: Is it possible to get a higher resolution figure here?

This is a schematic of how these packages work together to build a machine learning algorithm:

Benefits of tidymodels

The two major benefits of tidymodels are:

1. Standardized workflow/format/notation across different types of machine learning algorithms

Different notations are required for different algorithms as the algorithms have been developed by different people. This would require the painstaking process of reformatting the data to be compatible with each algorithm if multiple algorithms were tested.

2. Can easily modify pre-processing, algorithm choice, and hyper-parameter tuning making optimization easy

Modifying a piece of the overall process is now easier than before because many of the steps are specified using the tidymodels packages in a convenient manner. Thus the entire process can be rerun after a simple change to pre-processing without much difficulty.

Splitting the data

The first step after data exploration in machine learning analysis is to split the data into training and testing data sets.

The training data set will be used to build and tune our model. This is the data that the model “learns” on. The testing data set will be used to evaluate the performance of our model in a more generalizable way. What do we mean by “generalizable”?

Remember that our main goal is to use our model to be able to predict air pollution levels in areas where there are no gravimetric monitors. Therefore, if our model is really good at predicting air pollution with the data that we use to build it, it might not do the best job for the areas where there are few to no monitors. This would cause us to have really good prediction accuracy and we might assume that we were going to do a good job estimating air pollution any time we use our model, but in fact this would likely not be the case. This situation is what we call overfitting .

Overfitting happens when we end up modeling not only the major relationships in our data but also the noise within our data.

[source]

If we get good prediction with our testing set, then we know that our model can be applied to other data and will likely perform well. We will discuss this more later.

We will not touch the testing set until we have completed optimizing our model with the training set. This will allow us to have a less biased evaluation of how well our model can do with other data besides the data used in the training set to build the model. Ideally, you would also want a completely independent data set to further test the performance of your model.

To split the data into training and testing, we will use the initial_split() function in the rsample package to specify how we want to split our data.

AVOCADO: should we use “sample” or “observation”? Also, should we use “predictor” or “feature” in the figure? Could the figure be made to be more horizontal where the yellow and blue boxes are on top of each other?

set.seed(1234)
pm_split <- rsample::initial_split(data = pm, prop = 2/3)
pm_split

<Analysis/Assess/Total>
<584/292/876>

A couple of notes from the code above:

• Typically, data are split into 3/4 of the observations for training and 1/4 for testing. This is the default proportion and does not need to be specified. However, you can change the proportion using the prop argument, which we will do that here for illustrative purposes.
• Since the split is performed randomly, it is a good idea to use the set.seed() function in base R to ensure that if your rerun your code that your split will be the same next time.
• We can see the number of monitors in our training, testing, and original data by typing in the name of our split object. The result will look like this: <training data sample number, testing data sample number, original sample number>

Now, you can also specify a variable to stratify by with the strata argument. This is useful if you have imbalanced categorical variables and you would like to intentionally make sure that there are similar number of samples of the rarer categories in both the testing and training sets. Otherwise the split is performed randomly.

AVOCADO: Is this a quote from someone/some place? If so should we add a citation / URL?

The strata argument causes the random sampling to be conducted within the stratification variable. The can help ensure that the number of data points in the training data is equivalent to the proportions in the original data set.

In the case with our data set, perhaps we would like our training set to have similar proportions of monitors from each of the states as in the initial data. This might be useful if we want our model to be generalizable across all of the states.

We can see that indeed there are different proportions of monitors in each state by using the count() function of the dplyr package.

count(pm, state)

Scroll through the output:

# A tibble: 49 x 2
   state                    n
   <chr>                <int>
 1 Alabama                 24
 2 Arizona                 17
 3 Arkansas                16
 4 California              85
 5 Colorado                15
 6 Connecticut             14
 7 Delaware                 7
 8 District Of Columbia     3
 9 Florida                 29
10 Georgia                 28
11 Idaho                    7
12 Illinois                38
13 Indiana                 36
14 Iowa                    20
15 Kansas                  10
16 Kentucky                22
17 Louisiana               17
18 Maine                    1
19 Maryland                15
20 Massachusetts           16
21 Michigan                30
22 Minnesota               17
23 Mississippi             12
24 Missouri                13
25 Montana                 16
26 Nebraska                 7
27 Nevada                   4
28 New Hampshire            7
29 New Jersey              23
30 New Mexico              10
31 New York                24
32 North Carolina          35
33 North Dakota             4
34 Ohio                    44
35 Oklahoma                10
36 Oregon                  17
37 Pennsylvania            32
38 Rhode Island             5
39 South Carolina          14
40 South Dakota             9
41 Tennessee                3
42 Texas                   27
43 Utah                    14
44 Vermont                  4
45 Virginia                20
46 Washington               8
47 West Virginia           14
48 Wisconsin               21
49 Wyoming                 12

If our data set were large enough it might be nice then to stratify by state using the strata = "state" argument in initial_split(), but our data is unfortunately not large enough.

Importantly the initial_split function only determines what rows of our pm data frame should be assigned for training or testing, it does not actually split the data.

To extract the testing and training data we can use the training() and testing() functions also of the rsample package.

train_pm <-rsample::training(pm_split)
test_pm <-rsample::testing(pm_split)
 
# Scroll through the output!
count(train_pm, state)

# A tibble: 48 x 2
   state                    n
   <chr>                <int>
 1 Alabama                 18
 2 Arizona                 12
 3 Arkansas                14
 4 California              54
 5 Colorado                12
 6 Connecticut              8
 7 Delaware                 6
 8 District Of Columbia     2
 9 Florida                 18
10 Georgia                 17
# … with 38 more rows

count(test_pm, state)

# A tibble: 48 x 2
   state                    n
   <chr>                <int>
 1 Alabama                  6
 2 Arizona                  5
 3 Arkansas                 2
 4 California              31
 5 Colorado                 3
 6 Connecticut              6
 7 Delaware                 1
 8 District Of Columbia     1
 9 Florida                 11
10 Georgia                 11
# … with 38 more rows

Pre-processing the data

After splitting the data, the next step is to process the training and testing data so that the data are are compatible and optimized to be used with the model. This involves assigning variables to specific roles within the model and pre-processing like scaling variables and removing redundant variables. This process can also be called feature engineering.

To do this in tidymodels, we will create a what’s called a “recipe” using the recipes package, which is a standardized format for a sequence of steps for processing the data. This can be very useful because it makes testing out different pre-processing steps or different algorithms with the same pre-processing very easy and reproducible. Creating a recipe specifies how a data frame of predictors should be created - it specifies what variables to be used and the pre-processing steps, but it does not execute these steps or create the data frame of predictors.

Step 1: Specify the variables with recipe() function

The first thing to do to create a recipe is to specify which variables we will be using as our outcome and predictors using the recipe() function. In terms of the metaphor of baking, we can think of this as listing our ingredients. Translating this to the recipes package, we use the recipe() function to assign roles to all the variables.

Let’s try the simplest recipe with no pre-processing steps: simply list the outcome and predictor variables.

We can do so in two ways:

1. Using formula notation
   
2. Assigning roles to each variable

Let’s look at the first way using formula notation, which looks like this:

outcome(s) ~ predictor(s)

If in the case of multiple predictors or a multivariate situation with two outcomes, use a plus sign

outcome1 + outcome2 ~ predictor1 + predictor2

If we want to include all predictors we can use a period like so:

outcome_variable_name ~ .

Now with our data, we will start by making a recipe for our training data. If you recall, the continuous outcome variable is value (the average annual gravimetric monitor PM_2.5 concentration in ug/m³). Our features (or predictor variables) are all the other variables except the monitor ID, which is an id variable.

The reason not to include the id variable is because this variable includes the county number and a number designating which particular monitor the values came from of the monitors there are in that county. Since this number is arbitrary and the county information is also given in the data, and the fact that each monitor only has one value in the value variable, nothing is gained by including this variable and it may instead introduce noise. However, it is useful to keep this data to take a look at what is happening later. We will show you what to do in this case in just a bit.

In the simplest case, we might use all predictors like this:

simple_rec <- train_pm %>%
  recipes::recipe(value ~ .)

simple_rec

Data Recipe

Inputs:

      role #variables
   outcome          1
 predictor         49

We see a recipe has been created with 1 outcome variable and 49 predictor variables (or features). Also, notice how we named the output of recipe(). The naming convention for recipe objects is *_rec or rec.

Now, let’s get back to the id variable. Instead of including it as a predictor variable, we could also use the update_role() function of the recipes package.

simple_rec <- train_pm %>%
  recipes::recipe(value ~ .) %>%
  recipes::update_role(id, new_role = "id variable")

simple_rec

Data Recipe

Inputs:

        role #variables
 id variable          1
     outcome          1
   predictor         48

Click here learn more about the working with id variables

This option works well with the newer workflows package, however id variables are often dropped from analyses that do not use this newer package as they can make the process difficult with using the parsnip package alone due to the fact that new levels (or possible values) may be introduced with the testing data.

We could also specify the outcome and predictors in the same way as the id variable. Please see here for examples of other roles for variables. The role can be actually be any value.

The order is important here, as we first make all variables predictors and then override this role for the outcome and id variable. We will use the everything() function of the dplyr package to start with all of the variables in train_pm.

simple_rec <-recipe(train_pm) %>%
    update_role(everything(), new_role = "predictor")%>%
    update_role(value, new_role = "outcome")%>%
    update_role(id, new_role = "id variable")

simple_rec

Data Recipe

Inputs:

        role #variables
 id variable          1
     outcome          1
   predictor         48

If we want to take a look at our formula from our recipe, we can do use the formula() function of the stats package.

formula(simple_rec)

value ~ fips + lat + lon + state + county + city + CMAQ + zcta + 
    zcta_area + zcta_pop + imp_a500 + imp_a1000 + imp_a5000 + 
    imp_a10000 + imp_a15000 + county_area + county_pop + log_dist_to_prisec + 
    log_pri_length_5000 + log_pri_length_10000 + log_pri_length_15000 + 
    log_pri_length_25000 + log_prisec_length_500 + log_prisec_length_1000 + 
    log_prisec_length_5000 + log_prisec_length_10000 + log_prisec_length_15000 + 
    log_prisec_length_25000 + log_nei_2008_pm25_sum_10000 + log_nei_2008_pm25_sum_15000 + 
    log_nei_2008_pm25_sum_25000 + log_nei_2008_pm10_sum_10000 + 
    log_nei_2008_pm10_sum_15000 + log_nei_2008_pm10_sum_25000 + 
    popdens_county + popdens_zcta + nohs + somehs + hs + somecollege + 
    associate + bachelor + grad + pov + hs_orless + urc2013 + 
    urc2006 + aod
<environment: 0x7ffeb8aa5fd8>

We can also view our recipe in more detail using the base summary() function.

summary(simple_rec)

# A tibble: 50 x 4
   variable type    role        source  
   <chr>    <chr>   <chr>       <chr>   
 1 id       nominal id variable original
 2 value    numeric outcome     original
 3 fips     nominal predictor   original
 4 lat      numeric predictor   original
 5 lon      numeric predictor   original
 6 state    nominal predictor   original
 7 county   nominal predictor   original
 8 city     nominal predictor   original
 9 CMAQ     numeric predictor   original
10 zcta     nominal predictor   original
# … with 40 more rows

To summarize this step, we use the recipe() function to assign roles to all the variables:

Step 2: Specify the pre-processing steps with step*() functions

Next, we use the step*() functions from the recipe package to specify pre-processing steps.

This link and this link show the many options for recipe step functions.

There are step functions for a variety of purposes:

1. Imputation – filling in missing values based on the existing data
2. Transformation – changing all values of a variable in the same way, typically to make it more normal or easier to interpret
3. Discretization – converting continuous values into discrete or nominal values - binning for example to reduce the number of possible levels (However this is generally not advisable!)
4. Encoding / Creating Dummy Variables – creating a numeric code for categorical variables (More on Dummy Variables and one hot encoding)
5. Data type conversions – which means changing from integer to factor or numeric to date etc.
6. Interaction term addition to the model – which means that we would be modeling for predictors that would influence the capacity of each other to predict the outcome
7. Normalization – centering and scaling the data to a similar range of values
8. Dimensionality Reduction/ Signal Extraction – reducing the space of features or predictors to a smaller set of variables that capture the variation or signal in the original variables (ex. Principal Component Analysis and Independent Component Analysis)
9. Filtering – filtering options for removing variables (ex. remove variables that are highly correlated to others or remove variables with very little variance and therefore likely little predictive capacity)
10. Row operations – performing functions on the values within the rows (ex. rearranging, filtering, imputing)
11. Checking functions – Sanity checks to look for missing values, to look at the variable classes etc.

All of the step functions look like step_*() with the * replaced with a name, except for the check functions which look like check_*().

There are several ways to select what variables to apply steps to:

1. Using tidyselect methods: contains(), matches(), starts_with(), ends_with(), everything(), num_range()
   
2. Using the type: all_nominal(), all_numeric() , has_type()
3. Using the role: all_predictors(), all_outcomes(), has_role()
4. Using the name - use the actual name of the variable/variables of interest

Let’s try adding some steps to our recipe.

We might consider log transforming our population and area variables (that aren’t densities) - let’s take a look at the range of these variables.

pm %>%
  select(matches("_pop|_area")) %>%
  map(range)

AVOCADO: should we explain map()? AVOCADO: unclear why this is commented out? The next sentence suggests it was executed but I don’t think the code works?

We can see that the range for each of these variables is quite large, we can log transform this data using the step_log() function of the recipes package.

We would also want to potentially one hot encode some of our categorical variables so that they can be used with certain algorithms. We can do this with the step_dummy() function and the one_hot = TRUE argument. One hot encoding means that we do not simply encode our categorical variables numerically, as our numeric assignments can be interpreted by algorithms as having a particular rank or order. Instead, binary variables made of 1s and 0s are used to arbitrarily assign a numeric value that has no apparent order.

simple_rec %>%
  step_dummy(state, county, city, zcta, one_hot = TRUE)

Data Recipe

Inputs:

        role #variables
 id variable          1
     outcome          1
   predictor         48

Operations:

Dummy variables from state, county, city, zcta

Our fips variable includes a numeric code for state and county - and therefore is essentially a proxy for county. Since we already have county, we will just use it and keep the fips ID as another ID variable.

We can remove the fips variable from the predictors using update_role() to make sure that the role is no longer "predictor". We can make the role anything we want actually, so we will keep it something identifiable.

simple_rec %>%
  update_role("fips", new_role = "county id")

Data Recipe

Inputs:

        role #variables
   county id          1
 id variable          1
     outcome          1
   predictor         47

We might also want to remove variables that appear to be redundant and are highly correlated with others, as we know from our exploratory data analysis that many of our variables are correlated with one another. We can do this using the step_corr() function.

simple_rec %>%
  step_corr(all_predictors(), - CMAQ, - aod)

Data Recipe

Inputs:

        role #variables
 id variable          1
     outcome          1
   predictor         48

Operations:

Correlation filter on all_predictors, -, CMAQ, -, aod

Notice, we don’t want to remove some of our variables, like the CMAQ and aod variables.

It is also a good idea to remove variables with near-zero variance, which can be done with the step_nzv() function. Variables have low variance if all the values are very similar, the values are very sparse, or if they are highly imbalanced.

simple_rec %>%
  step_nzv(all_predictors(), - CMAQ, - aod)

Data Recipe

Inputs:

        role #variables
 id variable          1
     outcome          1
   predictor         48

Operations:

Sparse, unbalanced variable filter on all_predictors, -, CMAQ, -, aod

Click here to learn about examples where you might have near-zero variance variables

1. Similar Values - If the population density was nearly the same for every zcta that contained a monitor, then knowing the population density near our monitor would contribute little to our model in assisting us to predict monitor air pollution values.
2. Sparse Data - If all of the monitors were in locations where the populations did not attend graduate school, then these values would mostly be zero, again this would do very little to help us distinguish our air pollution monitors.When many of the values are zero this is also called sparse data.
   
3. Imbalanced Data If nearly all of the monitors were located in one particular state, and all the others only had one monitor each, then the real predictive value would simply be in knowing if a monitor is located in that particular state or not. In this case we don’t want to remove our variable, we just want to simplify it.

See this blog post about why removing near-zero variance variables isn’t always a good idea if we think that a variable might be especially informative.

Let’s put all this together now.

Remember: it is important to add the steps to the recipe in an order that makes sense just like with a cooking recipe.

First, we are going to create numeric values for our categorical variables, then we will look at correlation and near-zero variance. We do not want to remove the CMAQ and aod variables, so we can make sure they are kept in the model by excluding them from those steps. If we specifically wanted to remove a predictor we could use step_rm().

simple_rec %<>%
  update_role("fips", new_role = "county id") %>%
  step_dummy(state, county, city, zcta, one_hot = TRUE) %>%
  step_corr(all_predictors(), - CMAQ, - aod)%>%
  step_nzv(all_predictors(), - CMAQ, - aod)
  
simple_rec

Data Recipe

Inputs:

        role #variables
   county id          1
 id variable          1
     outcome          1
   predictor         47

Operations:

Dummy variables from state, county, city, zcta
Correlation filter on all_predictors, -, CMAQ, -, aod
Sparse, unbalanced variable filter on all_predictors, -, CMAQ, -, aod

Running the pre-processing

Step 1: Update the recipe with training data using prep()

The next major function of the recipes package is prep(). This function updates the recipe object based on the training data. It estimates parameters (estimating the required quantities and statistics required by the steps for the variables) for pre-processing and updates the model terms, as some of the predictors may be removed, this allows the recipe to be ready to use on other data sets. It does not necessarily actually execute the pre-processing itself, however we will specify in argument for it to do this so that we can take a look at the pre-processed data.

AVOCADO: I’m not sure I get what “updates the model terms” in the paragraph above means. I didn’t think a model had been specified yet?

There are some important arguments to know about:

1. training - you must supply a training data set to estimate parameters for pre-processing operations (recipe steps) - this may already be included in your recipe - as is the case for us
2. fresh - if fresh=TRUE, - will retrain and estimate parameters for any previous steps that were already prepped if you add more steps to the recipe
3. verbose - if verbose=TRUE, shows the progress as the steps are evaluated and the size of the pre-processed training set
4. retain - if retain=TRUE, then the pre-processed training set will be saved within the recipe (as template). This is good if you are likely to add more steps and do not want to rerun the prep() on the previous steps. However this can make the recipe size large. This is necessary if you want to actually look at the pre-processed data.

Let’s try out the prep() function:

prepped_rec <- prep(simple_rec, verbose = TRUE, retain = TRUE )

oper 1 step dummy [training] 
oper 2 step corr [training] 
oper 3 step nzv [training] 
The retained training set is ~ 0.26 Mb  in memory.

names(prepped_rec)

[1] "var_info"       "term_info"      "steps"          "template"      
[5] "levels"         "retained"       "tr_info"        "orig_lvls"     
[9] "last_term_info"

There are also lots of useful things to checkout in the output of prep(). You can see:

1. the steps that were run
   
2. the variable info (var_info)
   
3. the model (term_info)
4. the new levels of the variables
5. the original levels of the variables (orig_lvls)
6. info about the training data set size and completeness (tr_info)

Note: You may see the prep.recipe() function in material that you read about the recipes package. This is referring to the prep() function of the recipes package.

Step 2: Extracting pre-processed training data using juice()

Since we retained our pre-processed training data (i.e. prep(retain=TRUE)), we can take a look at it like by using the juice() function of the recipes package like this:

Let’s juice!

# Scroll through the output!
juiced_train <- juice(prepped_rec)
glimpse(juiced_train)

Rows: 584
Columns: 36
$ id                          <fct> 1003.001, 1027.0001, 1033.1002, 1055.001,…
$ value                       <dbl> 9.597647, 10.800000, 11.212174, 12.375394…
$ fips                        <fct> 1003, 1027, 1033, 1055, 1069, 1073, 1073,…
$ lat                         <dbl> 30.49800, 33.28126, 34.75878, 33.99375, 3…
$ lon                         <dbl> -87.88141, -85.80218, -87.65056, -85.9910…
$ CMAQ                        <dbl> 8.098836, 9.766208, 9.402679, 9.241744, 9…
$ zcta_area                   <dbl> 190980522, 374132430, 16716984, 154069359…
$ zcta_pop                    <dbl> 27829, 5103, 9042, 20045, 30217, 9010, 16…
$ imp_a500                    <dbl> 0.01730104, 1.96972318, 19.17301038, 16.4…
$ imp_a15000                  <dbl> 1.4386207, 0.3359198, 5.2472094, 5.161210…
$ county_area                 <dbl> 4117521611, 1564252280, 1534877333, 13856…
$ county_pop                  <dbl> 182265, 13932, 54428, 104430, 101547, 658…
$ log_dist_to_prisec          <dbl> 4.648181, 7.219907, 5.760131, 5.261457, 7…
$ log_pri_length_5000         <dbl> 8.517193, 8.517193, 8.517193, 9.066563, 8…
$ log_pri_length_25000        <dbl> 11.32735, 10.12663, 10.15769, 12.01356, 1…
$ log_prisec_length_500       <dbl> 7.295356, 6.214608, 8.611945, 8.740680, 6…
$ log_prisec_length_1000      <dbl> 8.195119, 7.600902, 9.735569, 9.627898, 7…
$ log_prisec_length_5000      <dbl> 10.815042, 10.170878, 11.770407, 11.72888…
$ log_prisec_length_10000     <dbl> 11.886803, 11.405543, 12.840663, 12.76827…
$ log_nei_2008_pm10_sum_15000 <dbl> 2.26783411, 3.31111648, 6.70127741, 4.462…
$ log_nei_2008_pm10_sum_25000 <dbl> 5.628728, 3.311116, 7.148858, 4.678311, 3…
$ popdens_county              <dbl> 44.265706, 8.906492, 35.460814, 75.367038…
$ popdens_zcta                <dbl> 145.7164307, 13.6395554, 540.8870404, 130…
$ nohs                        <dbl> 3.3, 11.6, 7.3, 4.3, 5.8, 7.1, 2.7, 11.1,…
$ somehs                      <dbl> 4.9, 19.1, 15.8, 13.3, 11.6, 17.1, 6.6, 1…
$ hs                          <dbl> 25.1, 33.9, 30.6, 27.8, 29.8, 37.2, 30.7,…
$ somecollege                 <dbl> 19.7, 18.8, 20.9, 29.2, 21.4, 23.5, 25.7,…
$ associate                   <dbl> 8.2, 8.0, 7.6, 10.1, 7.9, 7.3, 8.0, 4.1, …
$ bachelor                    <dbl> 25.3, 5.5, 12.7, 10.0, 13.7, 5.9, 17.6, 7…
$ grad                        <dbl> 13.5, 3.1, 5.1, 5.4, 9.8, 2.0, 8.7, 2.9, …
$ pov                         <dbl> 6.1, 19.5, 19.0, 8.8, 15.6, 25.5, 7.3, 8.…
$ hs_orless                   <dbl> 33.3, 64.6, 53.7, 45.4, 47.2, 61.4, 40.0,…
$ urc2013                     <dbl> 4, 6, 4, 4, 4, 1, 1, 1, 1, 1, 2, 3, 3, 3,…
$ aod                         <dbl> 37.363636, 34.818182, 36.000000, 43.41666…
$ state_California            <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ city_Not.in.a.city          <dbl> 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0,…

For easy comparison sake - here is our original data:

# Scroll through the output!
glimpse(pm)

Rows: 876
Columns: 50
$ id                          <fct> 1003.001, 1027.0001, 1033.1002, 1049.1003…
$ value                       <dbl> 9.597647, 10.800000, 11.212174, 11.659091…
$ fips                        <fct> 1003, 1027, 1033, 1049, 1055, 1069, 1073,…
$ lat                         <dbl> 30.49800, 33.28126, 34.75878, 34.28763, 3…
$ lon                         <dbl> -87.88141, -85.80218, -87.65056, -85.9683…
$ state                       <chr> "Alabama", "Alabama", "Alabama", "Alabama…
$ county                      <chr> "Baldwin", "Clay", "Colbert", "DeKalb", "…
$ city                        <chr> "Fairhope", "Ashland", "Muscle Shoals", "…
$ CMAQ                        <dbl> 8.098836, 9.766208, 9.402679, 8.534772, 9…
$ zcta                        <fct> 36532, 36251, 35660, 35962, 35901, 36303,…
$ zcta_area                   <dbl> 190980522, 374132430, 16716984, 203836235…
$ zcta_pop                    <dbl> 27829, 5103, 9042, 8300, 20045, 30217, 90…
$ imp_a500                    <dbl> 0.01730104, 1.96972318, 19.17301038, 5.78…
$ imp_a1000                   <dbl> 1.4096021, 0.8531574, 11.1448962, 3.86764…
$ imp_a5000                   <dbl> 3.3360118, 0.9851479, 15.1786154, 1.23114…
$ imp_a10000                  <dbl> 1.9879187, 0.5208189, 9.7253870, 1.031646…
$ imp_a15000                  <dbl> 1.4386207, 0.3359198, 5.2472094, 0.973044…
$ county_area                 <dbl> 4117521611, 1564252280, 1534877333, 20126…
$ county_pop                  <dbl> 182265, 13932, 54428, 71109, 104430, 1015…
$ log_dist_to_prisec          <dbl> 4.648181, 7.219907, 5.760131, 3.721489, 5…
$ log_pri_length_5000         <dbl> 8.517193, 8.517193, 8.517193, 8.517193, 9…
$ log_pri_length_10000        <dbl> 9.210340, 9.210340, 9.274303, 10.409411, …
$ log_pri_length_15000        <dbl> 9.630228, 9.615805, 9.658899, 11.173626, …
$ log_pri_length_25000        <dbl> 11.32735, 10.12663, 10.15769, 11.90959, 1…
$ log_prisec_length_500       <dbl> 7.295356, 6.214608, 8.611945, 7.310155, 8…
$ log_prisec_length_1000      <dbl> 8.195119, 7.600902, 9.735569, 8.585843, 9…
$ log_prisec_length_5000      <dbl> 10.815042, 10.170878, 11.770407, 10.21420…
$ log_prisec_length_10000     <dbl> 11.88680, 11.40554, 12.84066, 11.50894, 1…
$ log_prisec_length_15000     <dbl> 12.205723, 12.042963, 13.282656, 12.35366…
$ log_prisec_length_25000     <dbl> 13.41395, 12.79980, 13.79973, 13.55979, 1…
$ log_nei_2008_pm25_sum_10000 <dbl> 0.318035438, 3.218632928, 6.573127301, 0.…
$ log_nei_2008_pm25_sum_15000 <dbl> 1.967358961, 3.218632928, 6.581917457, 3.…
$ log_nei_2008_pm25_sum_25000 <dbl> 5.067308, 3.218633, 6.875900, 4.887665, 4…
$ log_nei_2008_pm10_sum_10000 <dbl> 1.35588511, 3.31111648, 6.69187313, 0.000…
$ log_nei_2008_pm10_sum_15000 <dbl> 2.26783411, 3.31111648, 6.70127741, 3.350…
$ log_nei_2008_pm10_sum_25000 <dbl> 5.628728, 3.311116, 7.148858, 5.171920, 4…
$ popdens_county              <dbl> 44.265706, 8.906492, 35.460814, 35.330814…
$ popdens_zcta                <dbl> 145.716431, 13.639555, 540.887040, 40.718…
$ nohs                        <dbl> 3.3, 11.6, 7.3, 14.3, 4.3, 5.8, 7.1, 2.7,…
$ somehs                      <dbl> 4.9, 19.1, 15.8, 16.7, 13.3, 11.6, 17.1, …
$ hs                          <dbl> 25.1, 33.9, 30.6, 35.0, 27.8, 29.8, 37.2,…
$ somecollege                 <dbl> 19.7, 18.8, 20.9, 14.9, 29.2, 21.4, 23.5,…
$ associate                   <dbl> 8.2, 8.0, 7.6, 5.5, 10.1, 7.9, 7.3, 8.0, …
$ bachelor                    <dbl> 25.3, 5.5, 12.7, 7.9, 10.0, 13.7, 5.9, 17…
$ grad                        <dbl> 13.5, 3.1, 5.1, 5.8, 5.4, 9.8, 2.0, 8.7, …
$ pov                         <dbl> 6.1, 19.5, 19.0, 13.8, 8.8, 15.6, 25.5, 7…
$ hs_orless                   <dbl> 33.3, 64.6, 53.7, 66.0, 45.4, 47.2, 61.4,…
$ urc2013                     <dbl> 4, 6, 4, 6, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2,…
$ urc2006                     <dbl> 5, 6, 4, 5, 4, 4, 1, 1, 1, 1, 1, 1, 1, 2,…
$ aod                         <dbl> 37.36364, 34.81818, 36.00000, 33.08333, 4…

Notice how we only have 36 variables now instead of 50! Two of these are our ID variables (fips and the actual monitor ID (id)) and one is our outcome (value). Thus we only have 33 predictors now. We can also see that variables that we no longer have any categorical variables. Variables like state are gone and only state_California remains as it was the only state identity to have nonzero variance. We can also see that there were more monitors listed as "Not in a city" than any city.

We can see that California had the largest number of monitors compared to the other states.

pm %>% count(state) 

Scroll through the output:

# A tibble: 49 x 2
   state                    n
   <chr>                <int>
 1 Alabama                 24
 2 Arizona                 17
 3 Arkansas                16
 4 California              85
 5 Colorado                15
 6 Connecticut             14
 7 Delaware                 7
 8 District Of Columbia     3
 9 Florida                 29
10 Georgia                 28
11 Idaho                    7
12 Illinois                38
13 Indiana                 36
14 Iowa                    20
15 Kansas                  10
16 Kentucky                22
17 Louisiana               17
18 Maine                    1
19 Maryland                15
20 Massachusetts           16
21 Michigan                30
22 Minnesota               17
23 Mississippi             12
24 Missouri                13
25 Montana                 16
26 Nebraska                 7
27 Nevada                   4
28 New Hampshire            7
29 New Jersey              23
30 New Mexico              10
31 New York                24
32 North Carolina          35
33 North Dakota             4
34 Ohio                    44
35 Oklahoma                10
36 Oregon                  17
37 Pennsylvania            32
38 Rhode Island             5
39 South Carolina          14
40 South Dakota             9
41 Tennessee                3
42 Texas                   27
43 Utah                    14
44 Vermont                  4
45 Virginia                20
46 Washington               8
47 West Virginia           14
48 Wisconsin               21
49 Wyoming                 12

pm %>% count(city)

Scroll through the output:

# A tibble: 607 x 2
    city                                                 n
    <chr>                                            <int>
  1 Aberdeen                                             1
  2 Akron                                                2
  3 Albany                                               3
  4 Albuquerque                                          2
  5 Alexandria                                           1
  6 Allen Park                                           1
  7 Altamont                                             1
  8 Alton                                                1
  9 Amarillo                                             1
 10 Anadarko                                             1
 11 Anaheim                                              1
 12 Anderson                                             1
 13 Annandale                                            1
 14 Apache Junction                                      1
 15 Apple Valley                                         1
 16 Appleton                                             1
 17 Arden-Arcade                                         1
 18 Arlington                                            1
 19 Arnold                                               1
 20 Asheville                                            1
 21 Ashland                                              2
 22 Atascadero                                           1
 23 Athens-Clarke County (Remainder)                     1
 24 Atlanta                                              2
 25 Atlantic City                                        1
 26 Augusta-Richmond County (Remainder)                  2
 27 Aurora                                               1
 28 Austin                                               1
 29 Azusa                                                1
 30 Bakersfield                                          3
 31 Baltimore                                            5
 32 Batavia                                              1
 33 Baton Rouge                                          1
 34 Bay City                                             1
 35 Bayport                                              1
 36 Baytown                                              1
 37 Beaver Falls                                         1
 38 Beckley                                              1
 39 Belle Glade                                          1
 40 Bellevue                                             1
 41 Beltsville                                           1
 42 Bend                                                 1
 43 Bennington                                           1
 44 Bensley                                              1
 45 Big Bear City                                        1
 46 Billings                                             1
 47 Birmingham                                           2
 48 Bismarck                                             1
 49 Bladensburg                                          1
 50 Blair                                                1
 51 Blue Ash                                             1
 52 Blue Island                                          1
 53 Boise (corporate name Boise City)                    1
 54 Boone                                                1
 55 Boston                                               4
 56 Boulder                                              1
 57 Boulevard                                            1
 58 Bountiful                                            1
 59 Braidwood                                            1
 60 Brawley                                              1
 61 Bridgeport                                           1
 62 Brigham City                                         1
 63 Bristol                                              2
 64 Brockton                                             1
 65 Brook Park                                           1
 66 Brookings                                            1
 67 Brunswick                                            1
 68 Bryson City (RR name Bryson)                         1
 69 Buffalo                                              1
 70 Burbank                                              1
 71 Burlington                                           2
 72 Burns                                                1
 73 Butte-Silver Bow (Remainder)                         1
 74 Calexico                                             1
 75 Camden                                               1
 76 Candor                                               1
 77 Canton                                               2
 78 Carlisle                                             1
 79 Carlstadt                                            3
 80 Cary                                                 1
 81 Casa Grande                                          1
 82 Cedar Rapids                                         2
 83 Cedarhurst                                           1
 84 Central Point                                        1
 85 Chalmette                                            1
 86 Champaign                                            1
 87 Chapel Hill                                          1
 88 Charleroi                                            1
 89 Charleston                                           2
 90 Charlotte                                            2
 91 Chattanooga                                          1
 92 Chelmsford (Chelmsford Center)                       1
 93 Chester                                              2
 94 Cheyenne                                             1
 95 Chicago                                              5
 96 Chickasaw                                            1
 97 Chicopee                                             1
 98 Childersburg                                         1
 99 Chula Vista                                          1
100 Cicero                                               1
101 Cincinnati                                           3
102 Clairton                                             1
103 Claremont                                            1
104 Clarion                                              1
105 Clarksburg                                           1
106 Clearwater                                           1
107 Cleveland                                            6
108 Clinton                                              2
109 Clive                                                1
110 Clovis                                               1
111 Cockeysville                                         1
112 Cody                                                 1
113 Coloma                                               1
114 Colorado Springs                                     1
115 Columbia                                             1
116 Columbia Falls                                       1
117 Columbus                                             4
118 Columbus (Remainder)                                 3
119 Colusa                                               1
120 Commerce City                                        1
121 Concord                                              1
122 Conway                                               1
123 Corcoran                                             1
124 Cornwall                                             1
125 Corpus Christi                                       2
126 Cottage Grove                                        1
127 Cottonwood West                                      1
128 Council Bluffs                                       1
129 Covington                                            1
130 Crossett                                             1
131 Crossville                                           1
132 Dale                                                 1
133 Dallas                                               3
134 Danbury                                              1
135 Darrington                                           1
136 Davenport                                            3
137 Davie                                                1
138 Dayton                                               1
139 Dearborn                                             1
140 Decatur                                              2
141 Delray Beach                                         1
142 Dentsville (Dents)                                   1
143 Denver                                               3
144 Des Moines                                           1
145 Des Plaines                                          1
146 Detroit                                              5
147 Doraville                                            1
148 Dothan                                               1
149 Douglas                                              1
150 Dover                                                1
151 Duluth                                               2
152 Durham                                               1
153 East Chicago                                         1
154 East Farmingdale                                     1
155 East Hartford                                        2
156 East Highland Park                                   1
157 East Providence                                      1
158 East Ridge                                           1
159 East Saint Louis                                     1
160 East Syracuse                                        1
161 Edgewood                                             1
162 El Cajon                                             1
163 El Centro                                            1
164 El Dorado                                            1
165 El Paso                                              3
166 Elgin                                                1
167 Elizabeth                                            2
168 Elizabethtown                                        1
169 Elkhart                                              1
170 Emmetsburg                                           1
171 Erie                                                 1
172 Escondido                                            1
173 Essex                                                1
174 Eugene                                               2
175 Eureka                                               2
176 Evansville                                           3
177 Fairfield                                            1
178 Fairhope                                             1
179 Fairmont                                             1
180 Fall River                                           1
181 Farmington                                           1
182 Farrell                                              1
183 Fayetteville                                         1
184 Ferry Pass                                           1
185 Flagstaff                                            1
186 Flint                                                1
187 Follansbee                                           1
188 Fontana                                              1
189 Forest Park                                          1
190 Fort Collins                                         1
191 Fort Defiance                                        1
192 Fort Lee                                             1
193 Fort Myers                                           1
194 Fort Pierce                                          1
195 Fort Smith                                           1
196 Fort Wayne                                           1
197 Fort Worth                                           2
198 Frankfort                                            1
199 Franklin                                             1
200 Freemansburg                                         1
201 Fremont                                              1
202 Fresno                                               2
203 Gadsden                                              1
204 Gainesville                                          3
205 Galloway (Township of)                               1
206 Garden City                                          1
207 Gary                                                 3
208 Gastonia                                             1
209 Gilroy                                               1
210 Glen Burnie                                          1
211 Goldsboro                                            1
212 Gordon                                               1
213 Grand Island                                         1
214 Grand Junction                                       1
215 Grand Rapids                                         2
216 Granite City                                         2
217 Grants Pass                                          1
218 Grass Valley                                         1
219 Great Falls                                          1
220 Greater Upper Marlboro                               1
221 Greeley                                              1
222 Green Bay                                            1
223 Greensboro                                           1
224 Greensburg                                           1
225 Greenville                                           3
226 Greenwich (Township of)                              1
227 Grenada                                              1
228 Griffith                                             1
229 Groveton                                             1
230 Gulfport                                             1
231 Hamilton                                             1
232 Hammond                                              3
233 Hampton                                              1
234 Harrison Township                                    1
235 Harrisville                                          1
236 Hattiesburg                                          1
237 Haverhill                                            1
238 Helena                                               2
239 Helena Valley West Central                           1
240 Hernando                                             1
241 Hickory                                              1
242 Highland                                             1
243 Highland Heights                                     1
244 Hillsboro                                            1
245 Hobbs                                                1
246 Holland                                              1
247 Hollister                                            1
248 Hollywood                                            1
249 Homestead                                            1
250 Hoover                                               1
251 Hopewell (Township of)                               1
252 Hot Springs (Hot Springs National Park)              1
253 Houston                                              2
254 Huntington                                           1
255 Huntsville                                           1
256 Indianapolis                                         1
257 Indianapolis (Remainder)                             4
258 Indio                                                1
259 Iowa City                                            1
260 Ironton                                              1
261 Jackson                                              2
262 Jacksonville                                         2
263 Jamesville                                           1
264 Jasper                                               3
265 Jean                                                 1
266 Jeffersonville                                       1
267 Jenison                                              1
268 Jersey City                                          1
269 Jerseyville                                          1
270 Johnstown                                            1
271 Joliet                                               1
272 Kalamazoo                                            1
273 Kalispell                                            1
274 Kansas City                                          3
275 Keeler                                               1
276 Keene                                                1
277 Kenansville                                          1
278 Kenner                                               1
279 Kennesaw                                             1
280 Keokuk                                               1
281 Kinston                                              1
282 Kokomo                                               1
283 La Crosse                                            1
284 La Grande                                            1
285 Lackawanna                                           1
286 Laconia                                              1
287 Ladue                                                1
288 Lafayette                                            3
289 Lake Charles                                         1
290 Lakeland                                             1
291 Lakeport                                             1
292 Lakeview                                             1
293 Lancaster                                            2
294 Lander                                               1
295 Lansing                                              1
296 Las Cruces                                           1
297 Las Vegas                                            1
298 Laurel                                               1
299 Lawrence                                             1
300 Leander                                              1
301 Lebanon                                              2
302 Leeds                                                1
303 Lexington                                            1
304 Lexington-Fayette (corporate name for Lexington)     2
305 Libby                                                1
306 Liberty                                              1
307 Lincoln                                              1
308 Lindon                                               1
309 Little Rock                                          2
310 Littleton                                            1
311 Live Oak                                             1
312 Livermore                                            1
313 Livonia                                              1
314 Logan                                                1
315 Long Beach                                           2
316 Longmont                                             1
317 Los Angeles                                          1
318 Louisville                                           4
319 Luna Pier                                            1
320 Lynchburg                                            1
321 Lynn                                                 1
322 Lynwood                                              1
323 Macon                                                2
324 Madison                                              1
325 Magna                                                1
326 Mamaroneck                                           1
327 Manistee                                             1
328 Marble City Community                                1
329 Maricopa                                             1
330 Marion                                               2
331 Marrero                                              1
332 Martinsburg                                          1
333 Marysville                                           1
334 McAlester                                            1
335 McCook                                               1
336 McDonald                                             1
337 McLean                                               1
338 Medford                                              1
339 Melbourne                                            1
340 Mena                                                 1
341 Merced                                               1
342 Meridian                                             1
343 Mesa                                                 1
344 Miami                                                2
345 Michigan City                                        1
346 Middlesborough (corporate name for Middlesboro)      1
347 Middletown                                           2
348 Midlothian                                           1
349 Milwaukee                                            5
350 Mingo Junction                                       1
351 Minneapolis                                          2
352 Mira Loma                                            1
353 Mission                                              1
354 Mission Viejo                                        1
355 Missoula                                             1
356 Modesto                                              1
357 Mojave                                               1
358 Monroe                                               1
359 Montgomery                                           1
360 Morgantown                                           1
361 Morristown                                           1
362 Moundsville                                          1
363 Muncie                                               1
364 Muscatine                                            1
365 Muscle Shoals                                        1
366 Muskegon                                             1
367 Muskogee                                             1
368 Naperville                                           1
369 Nashua                                               1
370 Natchez                                              1
371 New Albany                                           1
372 New Haven                                            5
373 New Paris                                            1
374 New York                                             9
375 Newark                                               2
376 Newburgh                                             1
377 Newburgh Heights                                     1
378 Newport                                              1
379 Niagara Falls                                        1
380 Nogales                                              1
381 Norfolk                                              1
382 Normal                                               1
383 Norristown                                           1
384 North Braddock                                       1
385 North Brunswick Township                             1
386 North Charleston                                     1
387 North Las Vegas                                      1
388 North Little Rock                                    1
389 Northbrook                                           1
390 Norwalk                                              1
391 Norwich                                              1
392 Norwood                                              1
393 Not in a city                                      103
394 Oak Park                                             1
395 Oakland                                              2
396 Oakridge                                             1
397 Odessa                                               1
398 Ogden                                                1
399 Ogden Dunes (Wickliffe)                              1
400 Oglesby                                              1
401 Oklahoma City                                        2
402 Omaha                                                2
403 Onamia                                               1
404 Ontario                                              1
405 Orlando                                              1
406 Overland Park                                        1
407 Paducah                                              1
408 Painesville                                          1
409 Palm Springs                                         1
410 Palm Springs North                                   1
411 Panama City                                          1
412 Pasadena                                             1
413 Pascagoula                                           1
414 Paterson                                             1
415 Pawtucket                                            1
416 Peach Springs                                        1
417 Pelham                                               1
418 Pendleton                                            1
419 Pennsauken (Pensauken)                               1
420 Peoria                                               1
421 Phenix City                                          1
422 Philadelphia                                         5
423 Phillipsburg                                         1
424 Phoenix                                              3
425 Pico Rivera                                          1
426 Pikeville                                            1
427 Pinedale                                             1
428 Pinehurst (Pine Creek)                               1
429 Pinson                                               1
430 Piru                                                 1
431 Pittsboro                                            1
432 Pittsburgh                                           1
433 Pittsfield                                           1
434 Platteville                                          1
435 Pleasant Prairie                                     1
436 Pompano Beach                                        1
437 Port Arthur                                          1
438 Port Huron                                           1
439 Portland                                             2
440 Portola                                              1
441 Portsmouth                                           2
442 Potosi                                               1
443 Potsdam                                              1
444 Powder Springs                                       1
445 Prescott Valley                                      1
446 Presque Isle                                         1
447 Providence                                           2
448 Provo                                                1
449 Pryor (corporate name Pryor Creek)                   1
450 Pueblo                                               1
451 Quincy                                               2
452 Rahway                                               1
453 Raleigh                                              1
454 Rapid City                                           2
455 Ravenna                                              1
456 Redding                                              1
457 Redwood City                                         1
458 Reno                                                 1
459 Reseda                                               1
460 Richfield                                            1
461 Richmond                                             1
462 Ridge Wood Heights                                   1
463 Ridgecrest                                           1
464 Rio Rancho Estates                                   1
465 Riverside                                            1
466 Roanoke                                              2
467 Rochester                                            2
468 Rock Island Arsenal (U.S. Army)                      1
469 Rock Springs                                         1
470 Rockford                                             1
471 Rockwell                                             1
472 Rocky Mount                                          1
473 Rome                                                 1
474 Roseville                                            1
475 Roswell                                              1
476 Roxborough Park                                      1
477 Royal Palm Beach                                     1
478 Rubidoux                                             1
479 Russellville                                         1
480 Rutland                                              1
481 Sacramento                                           2
482 Saint Petersburg                                     1
483 Salinas                                              1
484 Salt Lake City                                       2
485 San Andreas                                          1
486 San Antonio                                          2
487 San Bernardino                                       1
488 San Diego                                            2
489 San Francisco                                        1
490 San Jose                                             1
491 San Luis Obispo                                      1
492 Sandersville                                         1
493 Sanford                                              1
494 Santa Barbara                                        1
495 Santa Fe                                             1
496 Santa Maria                                          1
497 Santa Rosa                                           1
498 Sault Ste. Marie                                     2
499 Savannah                                             1
500 Schiller Park                                        1
501 Scottsbluff                                          1
502 Scottsdale                                           1
503 Scranton                                             1
504 Seaford                                              1
505 Searcy                                               1
506 Seattle                                              2
507 Seeley Lake                                          1
508 Seven Oaks                                           1
509 Shakopee                                             1
510 Sharonville                                          1
511 Shasta Lake                                          1
512 Sheffield                                            1
513 Shepherdsville                                       1
514 Sheridan                                             2
515 Shreveport                                           1
516 Silver City                                          1
517 Simi Valley                                          1
518 Sioux City                                           1
519 Sioux Falls                                          2
520 Soddy-Daisy                                          1
521 South Bend                                           2
522 South Charleston                                     1
523 South Padre Island                                   1
524 Spanish Fork                                         1
525 Spokane                                              1
526 Springdale                                           1
527 Springfield                                          5
528 Spruce Pine                                          1
529 St. Bernard                                          1
530 St. Cloud                                            1
531 St. Joseph                                           1
532 St. Louis                                            4
533 St. Louis Park                                       1
534 St. Paul                                             3
535 State College                                        1
536 Ste. Genevieve                                       1
537 Steubenville                                         1
538 Stockton                                             1
539 Stuttgart                                            1
540 Summit                                               1
541 Suncook                                              1
542 Swansea                                              1
543 Tacoma                                               1
544 Tallahassee                                          1
545 Tampa                                                1
546 Taylors                                              1
547 Tecumseh                                             1
548 Terre Haute                                          2
549 Texarkana                                            1
550 Theodore                                             1
551 Thomaston                                            1
552 Thompson Falls                                       1
553 Thousand Oaks                                        1
554 Toledo                                               3
555 Toms River                                           1
556 Tooele                                               1
557 Topeka                                               1
558 Trenton                                              1
559 Truckee                                              1
560 Tucson                                               2
561 Tulsa                                                2
562 Tupelo                                               1
563 Tuscaloosa                                           1
564 Ukiah                                                1
565 Underhill (Town of)                                  1
566 Union City                                           1
567 Valdosta                                             1
568 Vallejo                                              1
569 Valrico                                              1
570 Vancouver                                            1
571 Victorville                                          1
572 Vienna                                               1
573 Vinton                                               1
574 Virginia                                             1
575 Virginia Beach                                       1
576 Visalia                                              1
577 Warner Robins                                        1
578 Warren                                               1
579 Washington                                           4
580 Waterbury                                            1
581 Waterloo                                             1
582 Watertown                                            1
583 Waukesha                                             1
584 Waynesville                                          1
585 Weirton                                              2
586 West Orange                                          1
587 West Yellowstone                                     1
588 Westfield                                            1
589 Westport                                             1
590 Wheeling                                             1
591 Whitefish                                            1
592 Wichita                                              3
593 Wilmington                                           1
594 Winston-Salem                                        2
595 Winter Park                                          1
596 Wood River                                           1
597 Woodland                                             1
598 Worcester                                            2
599 Wyandotte                                            1
600 Yakima                                               1
601 Yellow Springs                                       1
602 York                                                 1
603 Youngstown                                           2
604 Ypsilanti                                            1
605 Yuba City                                            1
606 Yuma                                                 1
607 Zion                                                 1

Note: Recall that you must specify retain = TRUE argument of the prep() function to use juice().

Step 3: Extracting pre-processed testing data using bake()

According to the tidymodels documentation:

bake() takes a trained recipe and applies the operations to a data set to create a design matrix. For example: it applies the centering to new data sets using these means used to create the recipe

Therefore, if you wanted to look at the pre-processed testing data you would use the bake() function of the recipes package. (You generally want to leave your testing data alone, but it is good to look for issues like the introduction of NA values).

Let’s bake!

# Scroll through the output!
baked_test_pm <- recipes::bake(prepped_rec, new_data = test_pm)
glimpse(baked_test_pm)

Rows: 292
Columns: 36
$ id                          <fct> 1049.1003, 1073.101, 1073.2006, 1089.0014…
$ value                       <dbl> 11.659091, 13.114545, 12.228125, 12.23294…
$ fips                        <fct> 1049, 1073, 1073, 1089, 1103, 1121, 4013,…
$ lat                         <dbl> 34.28763, 33.54528, 33.38639, 34.68767, 3…
$ lon                         <dbl> -85.96830, -86.54917, -86.81667, -86.5863…
$ CMAQ                        <dbl> 8.534772, 9.303766, 10.235612, 9.343611, …
$ zcta_area                   <dbl> 203836235, 148994881, 56063756, 46963946,…
$ zcta_pop                    <dbl> 8300, 14212, 32390, 21297, 30545, 7713, 5…
$ imp_a500                    <dbl> 5.78200692, 0.06055363, 42.42820069, 23.2…
$ imp_a15000                  <dbl> 0.9730444, 2.9956557, 12.7487614, 10.3555…
$ county_area                 <dbl> 2012662359, 2878192209, 2878192209, 20761…
$ county_pop                  <dbl> 71109, 658466, 658466, 334811, 119490, 82…
$ log_dist_to_prisec          <dbl> 3.721489, 7.301545, 4.721755, 4.659519, 6…
$ log_pri_length_5000         <dbl> 8.517193, 9.683336, 10.737240, 8.517193, …
$ log_pri_length_25000        <dbl> 11.90959, 12.53777, 12.99669, 11.47391, 1…
$ log_prisec_length_500       <dbl> 7.310155, 6.214608, 7.528913, 8.760549, 6…
$ log_prisec_length_1000      <dbl> 8.585843, 7.600902, 9.342290, 9.543183, 8…
$ log_prisec_length_5000      <dbl> 10.214200, 11.262645, 11.713190, 11.48606…
$ log_prisec_length_10000     <dbl> 11.50894, 12.14101, 12.53899, 12.68440, 1…
$ log_nei_2008_pm10_sum_15000 <dbl> 3.3500444, 6.6241114, 5.8268686, 3.861625…
$ log_nei_2008_pm10_sum_25000 <dbl> 5.1719202, 7.5490587, 8.8205542, 5.219092…
$ popdens_county              <dbl> 35.330814, 228.777633, 228.777633, 161.26…
$ popdens_zcta                <dbl> 40.718962, 95.385827, 577.735106, 453.475…
$ nohs                        <dbl> 14.3, 7.2, 0.8, 1.2, 4.8, 16.7, 19.1, 6.4…
$ somehs                      <dbl> 16.7, 12.2, 2.6, 3.1, 7.8, 33.3, 15.6, 9.…
$ hs                          <dbl> 35.0, 32.2, 12.9, 15.1, 28.7, 37.5, 26.5,…
$ somecollege                 <dbl> 14.9, 19.0, 17.9, 20.5, 25.0, 12.5, 18.0,…
$ associate                   <dbl> 5.5, 6.8, 5.2, 6.5, 7.5, 0.0, 6.0, 8.8, 3…
$ bachelor                    <dbl> 7.9, 14.8, 35.5, 30.4, 18.2, 0.0, 10.6, 1…
$ grad                        <dbl> 5.8, 7.7, 25.2, 23.3, 8.0, 0.0, 4.1, 5.7,…
$ pov                         <dbl> 13.8, 10.5, 2.1, 5.2, 8.3, 18.8, 21.4, 14…
$ hs_orless                   <dbl> 66.0, 51.6, 16.3, 19.4, 41.3, 87.5, 61.2,…
$ urc2013                     <dbl> 6, 1, 1, 3, 4, 5, 1, 2, 5, 4, 4, 6, 6, 1,…
$ aod                         <dbl> 33.08333, 42.45455, 44.25000, 42.41667, 4…
$ state_California            <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,…
$ city_Not.in.a.city          <dbl> NA, NA, NA, NA, NA, NA, 0, NA, NA, NA, NA…

Notice that our city_Not.in.a.city variable seems to be NA values. Why might that be?

Ah! Perhaps it is because some of our levels were not previously seen in the training set!

Let’s take a look using the set operations of the dplyr package. We can take a look at cities that were different between the test and training set.

traincities <- train_pm %>% distinct(city)
testcities <- test_pm %>% distinct(city)

#get the number of cities that were different
dim(dplyr::setdiff(traincities, testcities))

[1] 376   1

#get the number of cities that overlapped
dim(dplyr::intersect(traincities, testcities))

[1] 51  1

Indeed, there are lots of different cities in our test data that are not in our training data!

AVOCADO: yeah, I would remove the paragraph below.

Maybe remove this?: Thus we need to update our original recipe to include a very important step function called step_novel() this helps in cases like this were there are new factors in our testing set that were not in our training set. It is a good idea to include this in most of your recipes where you have a categorical variables with many distinct values. This step needs to come before we create dummy variables. However, we are also creating a dummy variable from this, which still results in a problem.

Next, let go back to our pm data set and modify the city variable to be values of in a city or not in a city using the case_when() function of dplyr. This function allows you to vectorize multiple if_else() statements.

pm %>%
  mutate(city = case_when(city == "Not in a city" ~ "Not in a city",
                          city != "Not in a city" ~ "In a city"))

# A tibble: 876 x 50
   id    value fips    lat   lon state county city   CMAQ zcta  zcta_area
   <fct> <dbl> <fct> <dbl> <dbl> <chr> <chr>  <chr> <dbl> <fct>     <dbl>
 1 1003…  9.60 1003   30.5 -87.9 Alab… Baldw… In a…  8.10 36532 190980522
 2 1027… 10.8  1027   33.3 -85.8 Alab… Clay   In a…  9.77 36251 374132430
 3 1033… 11.2  1033   34.8 -87.7 Alab… Colbe… In a…  9.40 35660  16716984
 4 1049… 11.7  1049   34.3 -86.0 Alab… DeKalb In a…  8.53 35962 203836235
 5 1055… 12.4  1055   34.0 -86.0 Alab… Etowah In a…  9.24 35901 154069359
 6 1069… 10.5  1069   31.2 -85.4 Alab… Houst… In a…  9.12 36303 162685124
 7 1073… 15.6  1073   33.6 -86.8 Alab… Jeffe… In a… 10.2  35207  26929603
 8 1073… 12.4  1073   33.3 -87.0 Alab… Jeffe… Not … 10.2  35111 166239542
 9 1073… 11.1  1073   33.5 -87.3 Alab… Jeffe… Not …  8.16 35444 385566685
10 1073… 13.1  1073   33.5 -86.5 Alab… Jeffe… In a…  9.30 35094 148994881
# … with 866 more rows, and 39 more variables: zcta_pop <dbl>, imp_a500 <dbl>,
#   imp_a1000 <dbl>, imp_a5000 <dbl>, imp_a10000 <dbl>, imp_a15000 <dbl>,
#   county_area <dbl>, county_pop <dbl>, log_dist_to_prisec <dbl>,
#   log_pri_length_5000 <dbl>, log_pri_length_10000 <dbl>,
#   log_pri_length_15000 <dbl>, log_pri_length_25000 <dbl>,
#   log_prisec_length_500 <dbl>, log_prisec_length_1000 <dbl>,
#   log_prisec_length_5000 <dbl>, log_prisec_length_10000 <dbl>,
#   log_prisec_length_15000 <dbl>, log_prisec_length_25000 <dbl>,
#   log_nei_2008_pm25_sum_10000 <dbl>, log_nei_2008_pm25_sum_15000 <dbl>,
#   log_nei_2008_pm25_sum_25000 <dbl>, log_nei_2008_pm10_sum_10000 <dbl>,
#   log_nei_2008_pm10_sum_15000 <dbl>, log_nei_2008_pm10_sum_25000 <dbl>,
#   popdens_county <dbl>, popdens_zcta <dbl>, nohs <dbl>, somehs <dbl>,
#   hs <dbl>, somecollege <dbl>, associate <dbl>, bachelor <dbl>, grad <dbl>,
#   pov <dbl>, hs_orless <dbl>, urc2013 <dbl>, urc2006 <dbl>, aod <dbl>

Alternatively you could create a custom step function to do this and add the step function to your recipe, but that is beyond the scope of this case study.

We will need to repeat all the steps (splitting the data, pre-processing, etc) as the levels of our variables have now changed.

While we are doing this, we might also have this issue for state and county. So let’s also do a similar thing for state. The county variables appears to get dropped due to either correlation or near zero variance. It is likely due to near zero variance because this is the more granular of these geographic categorical variables and likely sparse.

pm %<>%
  mutate(city = case_when(city == "Not in a city" ~ "Not in a city",
                          city != "Not in a city" ~ "In a city"))

set.seed(1234) # same seed as before
pm_split <-rsample::initial_split(data = pm, prop = 2/3)
pm_split

<Analysis/Assess/Total>
<584/292/876>

 train_pm <-rsample::training(pm_split)
 test_pm <-rsample::testing(pm_split)

  
novel_rec <-recipe(train_pm) %>%
    update_role(everything(), new_role = "predictor") %>%
    update_role(value, new_role = "outcome") %>%
    update_role(id, new_role = "id variable") %>%
    update_role("fips", new_role = "county id") %>%
    step_dummy(state, county, city, zcta, one_hot = TRUE) %>%
    step_corr(all_numeric()) %>%
    step_nzv(all_numeric()) 

Now let’s retrain our training data and try baking our test data:

prepped_rec <- prep(novel_rec, verbose = TRUE, retain = TRUE)

oper 1 step dummy [training] 
oper 2 step corr [training] 
oper 3 step nzv [training] 
The retained training set is ~ 0.26 Mb  in memory.

# Scroll through the output!
juiced_train <- juice(prepped_rec)

glimpse(juiced_train)

Rows: 584
Columns: 36
$ id                          <fct> 1003.001, 1027.0001, 1033.1002, 1055.001,…
$ value                       <dbl> 9.597647, 10.800000, 11.212174, 12.375394…
$ fips                        <fct> 1003, 1027, 1033, 1055, 1069, 1073, 1073,…
$ lat                         <dbl> 30.49800, 33.28126, 34.75878, 33.99375, 3…
$ lon                         <dbl> -87.88141, -85.80218, -87.65056, -85.9910…
$ CMAQ                        <dbl> 8.098836, 9.766208, 9.402679, 9.241744, 9…
$ zcta_area                   <dbl> 190980522, 374132430, 16716984, 154069359…
$ zcta_pop                    <dbl> 27829, 5103, 9042, 20045, 30217, 9010, 16…
$ imp_a500                    <dbl> 0.01730104, 1.96972318, 19.17301038, 16.4…
$ imp_a15000                  <dbl> 1.4386207, 0.3359198, 5.2472094, 5.161210…
$ county_area                 <dbl> 4117521611, 1564252280, 1534877333, 13856…
$ county_pop                  <dbl> 182265, 13932, 54428, 104430, 101547, 658…
$ log_dist_to_prisec          <dbl> 4.648181, 7.219907, 5.760131, 5.261457, 7…
$ log_pri_length_5000         <dbl> 8.517193, 8.517193, 8.517193, 9.066563, 8…
$ log_pri_length_25000        <dbl> 11.32735, 10.12663, 10.15769, 12.01356, 1…
$ log_prisec_length_500       <dbl> 7.295356, 6.214608, 8.611945, 8.740680, 6…
$ log_prisec_length_1000      <dbl> 8.195119, 7.600902, 9.735569, 9.627898, 7…
$ log_prisec_length_5000      <dbl> 10.815042, 10.170878, 11.770407, 11.72888…
$ log_prisec_length_10000     <dbl> 11.886803, 11.405543, 12.840663, 12.76827…
$ log_nei_2008_pm10_sum_15000 <dbl> 2.26783411, 3.31111648, 6.70127741, 4.462…
$ log_nei_2008_pm10_sum_25000 <dbl> 5.628728, 3.311116, 7.148858, 4.678311, 3…
$ popdens_county              <dbl> 44.265706, 8.906492, 35.460814, 75.367038…
$ popdens_zcta                <dbl> 145.7164307, 13.6395554, 540.8870404, 130…
$ nohs                        <dbl> 3.3, 11.6, 7.3, 4.3, 5.8, 7.1, 2.7, 11.1,…
$ somehs                      <dbl> 4.9, 19.1, 15.8, 13.3, 11.6, 17.1, 6.6, 1…
$ hs                          <dbl> 25.1, 33.9, 30.6, 27.8, 29.8, 37.2, 30.7,…
$ somecollege                 <dbl> 19.7, 18.8, 20.9, 29.2, 21.4, 23.5, 25.7,…
$ associate                   <dbl> 8.2, 8.0, 7.6, 10.1, 7.9, 7.3, 8.0, 4.1, …
$ bachelor                    <dbl> 25.3, 5.5, 12.7, 10.0, 13.7, 5.9, 17.6, 7…
$ grad                        <dbl> 13.5, 3.1, 5.1, 5.4, 9.8, 2.0, 8.7, 2.9, …
$ pov                         <dbl> 6.1, 19.5, 19.0, 8.8, 15.6, 25.5, 7.3, 8.…
$ hs_orless                   <dbl> 33.3, 64.6, 53.7, 45.4, 47.2, 61.4, 40.0,…
$ urc2013                     <dbl> 4, 6, 4, 4, 4, 1, 1, 1, 1, 1, 2, 3, 3, 3,…
$ aod                         <dbl> 37.363636, 34.818182, 36.000000, 43.41666…
$ state_California            <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ city_Not.in.a.city          <dbl> 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0,…

Notice, it looks like we gained the log_prisec_length_25000 back with this recipe using the data with our changes to state and city.

# Scroll through the output!
baked_test_pm <- recipes::bake(prepped_rec, new_data = test_pm)

glimpse(baked_test_pm)

Rows: 292
Columns: 36
$ id                          <fct> 1049.1003, 1073.101, 1073.2006, 1089.0014…
$ value                       <dbl> 11.659091, 13.114545, 12.228125, 12.23294…
$ fips                        <fct> 1049, 1073, 1073, 1089, 1103, 1121, 4013,…
$ lat                         <dbl> 34.28763, 33.54528, 33.38639, 34.68767, 3…
$ lon                         <dbl> -85.96830, -86.54917, -86.81667, -86.5863…
$ CMAQ                        <dbl> 8.534772, 9.303766, 10.235612, 9.343611, …
$ zcta_area                   <dbl> 203836235, 148994881, 56063756, 46963946,…
$ zcta_pop                    <dbl> 8300, 14212, 32390, 21297, 30545, 7713, 5…
$ imp_a500                    <dbl> 5.78200692, 0.06055363, 42.42820069, 23.2…
$ imp_a15000                  <dbl> 0.9730444, 2.9956557, 12.7487614, 10.3555…
$ county_area                 <dbl> 2012662359, 2878192209, 2878192209, 20761…
$ county_pop                  <dbl> 71109, 658466, 658466, 334811, 119490, 82…
$ log_dist_to_prisec          <dbl> 3.721489, 7.301545, 4.721755, 4.659519, 6…
$ log_pri_length_5000         <dbl> 8.517193, 9.683336, 10.737240, 8.517193, …
$ log_pri_length_25000        <dbl> 11.90959, 12.53777, 12.99669, 11.47391, 1…
$ log_prisec_length_500       <dbl> 7.310155, 6.214608, 7.528913, 8.760549, 6…
$ log_prisec_length_1000      <dbl> 8.585843, 7.600902, 9.342290, 9.543183, 8…
$ log_prisec_length_5000      <dbl> 10.214200, 11.262645, 11.713190, 11.48606…
$ log_prisec_length_10000     <dbl> 11.50894, 12.14101, 12.53899, 12.68440, 1…
$ log_nei_2008_pm10_sum_15000 <dbl> 3.3500444, 6.6241114, 5.8268686, 3.861625…
$ log_nei_2008_pm10_sum_25000 <dbl> 5.1719202, 7.5490587, 8.8205542, 5.219092…
$ popdens_county              <dbl> 35.330814, 228.777633, 228.777633, 161.26…
$ popdens_zcta                <dbl> 40.718962, 95.385827, 577.735106, 453.475…
$ nohs                        <dbl> 14.3, 7.2, 0.8, 1.2, 4.8, 16.7, 19.1, 6.4…
$ somehs                      <dbl> 16.7, 12.2, 2.6, 3.1, 7.8, 33.3, 15.6, 9.…
$ hs                          <dbl> 35.0, 32.2, 12.9, 15.1, 28.7, 37.5, 26.5,…
$ somecollege                 <dbl> 14.9, 19.0, 17.9, 20.5, 25.0, 12.5, 18.0,…
$ associate                   <dbl> 5.5, 6.8, 5.2, 6.5, 7.5, 0.0, 6.0, 8.8, 3…
$ bachelor                    <dbl> 7.9, 14.8, 35.5, 30.4, 18.2, 0.0, 10.6, 1…
$ grad                        <dbl> 5.8, 7.7, 25.2, 23.3, 8.0, 0.0, 4.1, 5.7,…
$ pov                         <dbl> 13.8, 10.5, 2.1, 5.2, 8.3, 18.8, 21.4, 14…
$ hs_orless                   <dbl> 66.0, 51.6, 16.3, 19.4, 41.3, 87.5, 61.2,…
$ urc2013                     <dbl> 6, 1, 1, 3, 4, 5, 1, 2, 5, 4, 4, 6, 6, 1,…
$ aod                         <dbl> 33.08333, 42.45455, 44.25000, 42.41667, 4…
$ state_California            <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,…
$ city_Not.in.a.city          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…

Great now we no longer have NA values! :)

Note: if you use the skip option for some of the pre-processing steps, be careful. juice() will show all of the results ignoring skip = TRUE. bake() will not necessarily conduct these steps on the new data.

Specifying the model

So far we have used the packages rsample to split the data and recipes to assign variable types, and to specify and prep our pre-processing (as well as to optionally extract the pre-processed data).

We will now use the parsnip package (which is similar to the previous caret package - and hence why it is named after the vegetable) to specify our model.

There are four things we need to define about our model:

1. The type of model (using specific functions in parsnip like rand_forest(), logistic_reg() etc.)
   
2. The package or engine that we will use to implement the type of model selected (using the set_engine() function)
3. The mode of learning - classification or regression (using the set_mode() function)
4. Any arguments necessary for the model/package selected (using the set_args()function - for example the mtry = argument for random forest which is the number of variables to be used as options for splitting at each tree node)

Let’s walk through these steps one by one. For our case, we are going to start our analysis with a linear regression but we will demonstrate how we can try different models.

The first step is to define what type of model we would like to use. See here for modeling options in parsnip.

PM_model <- parsnip::linear_reg() # PM for particulate matter
PM_model

Linear Regression Model Specification (regression)

OK. So far, all we have defined is that we want to use a linear regression…
Let’s tell parsnip more about what we want.

We would like to use the ordinary least squares method to fit our linear regression. So we will tell parsnip that we want to use the lm package to implement our linear regression (there are many options actually such as rstan glmnet, keras, and sparklyr). We will do so by using the set_engine() function of the parsnip package.

lm_PM_model <- 
  PM_model  %>%
  parsnip::set_engine("lm")

lm_PM_model

Linear Regression Model Specification (regression)

Computational engine: lm 

In some cases some packages can do either classification or prediction, so it is a good idea to specify which mode you intend to perform. Here, we aim to predict the air pollution. You can do this with the set_mode() function of the parsnip package, by using either set_mode("classification") or set_mode("regression").

lm_PM_model <- 
  PM_model  %>%
  parsnip::set_engine("lm") %>%
  set_mode("regression")

lm_PM_model

Linear Regression Model Specification (regression)

Computational engine: lm 

Fitting the model

At the moment, there are actually a two ways to fit our model.

1. We can use the parsnip package and then assess our fit using the yardstick package.
2. We can also use the parsnip package with a newer package called workflows (still under development) that allows us to keep track of both our pre-processing steps and our model specification. It also allows us to implement fancier optimizations in an automated way and it is currently being developed to also handle post-processing operations.

AVOCADO: I feel like we should do a bit more explaining or maybe draw a figure of these two approaches?

Here we will talk about the second way to fit out model using the workflows and parsnip package.

We begin by creating a workflow using the workflow() function in the workflows package. Next, we use add_recipe() (our pre-processing specifications) and we add our model with the add_model() function – both functions from the workflows package.

Note: We do not need to actually prep() our recipe before using workflows!

If you recall novel_rec is the recipe we previously created with the recipes package and lm_PM_model is where we specified our model with the parsnip package. Here, we combine everything together into a workflow().

PM_wflow <-workflows::workflow() %>%
           workflows::add_recipe(novel_rec) %>%
           workflows::add_model(lm_PM_model)
PM_wflow

══ Workflow ═════════════════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: linear_reg()

── Preprocessor ─────────────────────────────────────────────────────────────────────────────
3 Recipe Steps

● step_dummy()
● step_corr()
● step_nzv()

── Model ────────────────────────────────────────────────────────────────────────────────────
Linear Regression Model Specification (regression)

Computational engine: lm 

Ah, nice. Notice how it tells us about both our pre-processing steps and our model specifications.

Next, we “prepare the recipe” (or estimate the parameters) and fit the model to our training data all at once. Printing the output, we can see the coefficients of the model.

PM_wflow_fit <- parsnip::fit(PM_wflow, data = train_pm)
PM_wflow_fit

══ Workflow [trained] ═══════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: linear_reg()

── Preprocessor ─────────────────────────────────────────────────────────────────────────────
3 Recipe Steps

● step_dummy()
● step_corr()
● step_nzv()

── Model ────────────────────────────────────────────────────────────────────────────────────

Call:
stats::lm(formula = formula, data = data)

Coefficients:
                (Intercept)                          lat  
                  2.221e+02                    3.773e-02  
                        lon                         CMAQ  
                  3.023e-02                    2.896e-01  
                  zcta_area                     zcta_pop  
                  3.592e-10                    7.988e-06  
                   imp_a500                   imp_a15000  
                  6.543e-03                   -1.252e-03  
                county_area                   county_pop  
                 -2.241e-11                   -2.052e-07  
         log_dist_to_prisec          log_pri_length_5000  
                  4.182e-02                   -2.207e-01  
       log_pri_length_25000        log_prisec_length_500  
                  9.277e-02                    2.651e-01  
     log_prisec_length_1000       log_prisec_length_5000  
                 -1.654e-02                    4.451e-01  
    log_prisec_length_10000  log_nei_2008_pm10_sum_15000  
                  1.510e-01                    1.029e-01  
log_nei_2008_pm10_sum_25000               popdens_county  
                  6.448e-02                   -3.905e-05  
               popdens_zcta                         nohs  
                 -2.098e-05                   -2.195e+00  
                     somehs                           hs  
                 -2.232e+00                   -2.235e+00  
                somecollege                    associate  
                 -2.237e+00                   -2.237e+00  
                   bachelor                         grad  
                 -2.239e+00                   -2.248e+00  
                        pov                    hs_orless  
                  5.400e-03                           NA  
                    urc2013                          aod  
                  1.848e-01                    2.521e-02  
           state_California           city_Not.in.a.city  
                  3.509e+00                    4.137e-01  

Click here learn how to do this without the workflows package

AVOCADO: this section feels like it should either have more explanation or just deleted. To me it feels confusing because it feels like we are showing a compeltely alternative path without an explanation why you might want to do? or when you would want to do this?

We could have also done this without the workflows package. Notice below that we use the processed training data (juiced_train) as opposed to the raw training data that we used with the workflow() we created with workflows above.

In this case, we actually need to write your model again! Recall that id and fips are ID variables and that values is our outcome of interest (the air pollution measure at each monitor).

juiced_train_ready <- juiced_train %>% 
  select(-id, -fips)

PM_fit <- lm_PM_model %>% 
  parsnip::fit(value ~., data = juiced_train_ready)

Assessing the model fit

After we fit our model, we can use the broom package allows for an easy/tidy way to look at the output from the fitted model.

The three functions from broom that we want to use are:

1. tidy() - returns a tidy data frame with coefficients from the model (one row per coefficient)
2. glance() - summarizes the model fit
3. augment() - shows the data, the fitted values (.fitted), the standard error (.se.fit) for each value and more

These broom functions currently only work with parsnip objects, not raw workflows objects. Here, we will use the PM_fit using parsnip.

broom::tidy(PM_fit) %>% 
  arrange(p.value)

# A tibble: 33 x 5
   term                 estimate    std.error statistic  p.value
   <chr>                   <dbl>        <dbl>     <dbl>    <dbl>
 1 state_California  3.51        0.433             8.11 3.25e-15
 2 CMAQ              0.290       0.0429            6.75 3.66e-11
 3 aod               0.0252      0.00574           4.39 1.37e- 5
 4 lon               0.0302      0.00931           3.25 1.23e- 3
 5 county_pop       -0.000000205 0.0000000931     -2.20 2.79e- 2
 6 urc2013           0.185       0.0980            1.89 5.98e- 2
 7 grad             -2.25        1.20             -1.87 6.20e- 2
 8 bachelor         -2.24        1.20             -1.86 6.28e- 2
 9 associate        -2.24        1.20             -1.86 6.29e- 2
10 somecollege      -2.24        1.20             -1.86 6.30e- 2
# … with 23 more rows

broom::glance(PM_fit[["fit"]])

# A tibble: 1 x 11
  r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
      <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
1     0.431         0.398  2.08      13.0 5.59e-49    33 -1241. 2549. 2698.
# … with 2 more variables: deviance <dbl>, df.residual <int>

broom::augment(PM_fit[["fit"]])  

# A tibble: 584 x 41
   value   lat   lon  CMAQ zcta_area zcta_pop imp_a500 imp_a15000 county_area
   <dbl> <dbl> <dbl> <dbl>     <dbl>    <dbl>    <dbl>      <dbl>       <dbl>
 1  9.60  30.5 -87.9  8.10 190980522    27829   0.0173      1.44   4117521611
 2 10.8   33.3 -85.8  9.77 374132430     5103   1.97        0.336  1564252280
 3 11.2   34.8 -87.7  9.40  16716984     9042  19.2         5.25   1534877333
 4 12.4   34.0 -86.0  9.24 154069359    20045  16.5         5.16   1385618994
 5 10.5   31.2 -85.4  9.12 162685124    30217  19.1         4.74   1501737720
 6 15.6   33.6 -86.8 10.2   26929603     9010  41.8        17.5    2878192209
 7 12.4   33.3 -87.0 10.2  166239542    16140   1.70        4.30   2878192209
 8 11.1   33.5 -87.3  8.16 385566685     3699   0           0.162  3423328940
 9 14.6   33.5 -86.9 10.2   10636977    11458  43.6        15.6    2878192209
10 12.0   33.7 -86.7  9.30 150661846    21725   1.48        4.25   2878192209
# … with 574 more rows, and 32 more variables: county_pop <dbl>,
#   log_dist_to_prisec <dbl>, log_pri_length_5000 <dbl>,
#   log_pri_length_25000 <dbl>, log_prisec_length_500 <dbl>,
#   log_prisec_length_1000 <dbl>, log_prisec_length_5000 <dbl>,
#   log_prisec_length_10000 <dbl>, log_nei_2008_pm10_sum_15000 <dbl>,
#   log_nei_2008_pm10_sum_25000 <dbl>, popdens_county <dbl>,
#   popdens_zcta <dbl>, nohs <dbl>, somehs <dbl>, hs <dbl>, somecollege <dbl>,
#   associate <dbl>, bachelor <dbl>, grad <dbl>, pov <dbl>, hs_orless <dbl>,
#   urc2013 <dbl>, aod <dbl>, state_California <dbl>, city_Not.in.a.city <dbl>,
#   .fitted <dbl>, .se.fit <dbl>, .resid <dbl>, .hat <dbl>, .sigma <dbl>,
#   .cooksd <dbl>, .std.resid <dbl>

However, we can use the broom functions (e.g. tidy()) with workflows, but we first need to use the pull_workflow_fit() function.

wflowoutput <- PM_wflow_fit %>% 
  pull_workflow_fit() %>% 
  broom::tidy() 

We can see that the output is identical using workflows to fit the model or using the output from parsnip

identical(tidy(PM_fit), wflowoutput)

[1] TRUE

We have fit our model on our training data, which means we have created a model to predict values of air pollution based on the predictors that we have included. Yay!

One last thing before we leave this section. We often are interested in getting a sense of which variables are the most important in our model. We can explore the variable importance using the vip() function of the vip package. This function create a bar plot of variable importance scores for each predictor variable (or feature) in a model. The bar plot is ordered by importance (highest to smallest).

Let’s take a look at the top 10 contributing variables:

PM_wflow_fit %>% 
  pull_workflow_fit() %>% 
  vip(num_features = 10)

Model performance

In this next section, our goal is to assess the overall model performance. The way we do this is to compare the similarity between the predicted estimates of the outcome variable produced by the model and the true outcome variable values.

If you recall the What is machine learning? section, we showed how to think about machine learning (ML) as an optimization problem that tries to minimize the distance between our predicted outcome \(\hat{Y} = f(X)\) and actual outcome \(Y\) using our features (or predictor variables) \(X\) as input to a function \(f\) that we want to estimate.

\[d(Y - \hat{Y})\]

As our goal in this section is to assess overall model performance, we will now talk about different distance metrics that you can use.

First, let’s pull out our predicted outcome values \(\hat{Y} = f(X)\) from the models we fit (using different approaches).

Using the parsnip package, we can grab the predicted outcome values (or fitted values) like this:

parsnip_fitted_values <- fitted(PM_fit[["fit"]])

head(parsnip_fitted_values)

        1         2         3         4         5         6 
 9.480088 10.418317 11.786217 11.295655 10.806653 11.091767 

Alternatively, we can get the fitted values using the augment() function using the output from parsnip:

parsnip_fitted_values <- 
  augment(PM_fit[["fit"]], data = juiced_train) %>% 
  select(value, .fitted:.std.resid)

head(parsnip_fitted_values)

# A tibble: 6 x 8
  value .fitted .se.fit .resid   .hat .sigma    .cooksd .std.resid
  <dbl>   <dbl>   <dbl>  <dbl>  <dbl>  <dbl>      <dbl>      <dbl>
1  9.60    9.48   0.375  0.118 0.0323   2.09 0.00000333     0.0573
2 10.8    10.4    0.383  0.382 0.0338   2.09 0.0000368      0.186 
3 11.2    11.8    0.404 -0.574 0.0376   2.09 0.0000933     -0.281 
4 12.4    11.3    0.366  1.08  0.0309   2.09 0.000267       0.526 
5 10.5    10.8    0.424 -0.298 0.0413   2.09 0.0000278     -0.146 
6 15.6    11.1    0.380  4.50  0.0332   2.08 0.00501        2.20  

Alternatively, we can use the output from workflows:

wf_fit <- PM_wflow_fit %>% 
  pull_workflow_fit()

wf_fitted_values <- fitted(wf_fit[["fit"]])
head(wf_fitted_values)

        1         2         3         4         5         6 
 9.480088 10.418317 11.786217 11.295655 10.806653 11.091767 

Alternatively, we can get the fitted values using the augment() function using the output from workflows:

wf_fitted_values <- 
  augment(wf_fit[["fit"]], data = juiced_train) %>% 
  select(value, .fitted:.std.resid)

head(wf_fitted_values)

# A tibble: 6 x 8
  value .fitted .se.fit .resid   .hat .sigma    .cooksd .std.resid
  <dbl>   <dbl>   <dbl>  <dbl>  <dbl>  <dbl>      <dbl>      <dbl>
1  9.60    9.48   0.375  0.118 0.0323   2.09 0.00000333     0.0573
2 10.8    10.4    0.383  0.382 0.0338   2.09 0.0000368      0.186 
3 11.2    11.8    0.404 -0.574 0.0376   2.09 0.0000933     -0.281 
4 12.4    11.3    0.366  1.08  0.0309   2.09 0.000267       0.526 
5 10.5    10.8    0.424 -0.298 0.0413   2.09 0.0000278     -0.146 
6 15.6    11.1    0.380  4.50  0.0332   2.08 0.00501        2.20  

Either way, this gives us the same fitted values

identical(parsnip_fitted_values, wf_fitted_values)

[1] TRUE

Visualizing model performance

Now, we can compare the predicted outcome values (or fitted values) \(\hat{Y}\) to the actual outcome values \(Y\) that we observed:

wf_fitted_values %>% 
  ggplot(aes(x =  value, y = .fitted)) + 
  geom_point() + 
  xlab("actual outcome values") + 
  ylab("predicted outcome values")

OK, so our range of the predicted outcome values appears to be smaller than the real values. We could probably do a bit better.

Quantifying model performance

Next, let’s use different distance functions \(d(\cdot)\) to assess how far off our predicted outcome \(\hat{Y} = f(X)\) and actual outcome \(Y\) values are from each other:

\[d(Y - \hat{Y})\]

As mentioned, there are entire scholarly fields of research dedicated to identifying different distance metrics \(d(\cdot)\) for machine learning applications. However, when performing prediction with a continuous outcome \(Y\), a few of the mostly commonly used distance metrics are:

1. mean absolute error (mae)

\[MAE = \frac{\sum_{i=1}^{n}{(|\hat{y_t}- y_t|)}^2}{n}\]

2. R squared error (rsq) – this is also known as the coefficient of determination which is the squared correlation between truth and estimate

This is calculated and 1 minus the fraction of the residual sum of squares (\(SS_res\)) by the total sum of squares (\(SS_tot\))

\[RSQ = R^2 = 1 - \frac{SSres}{SStot}\]

\[SS_{tot} = \sum_{i=1}^{n}{(y_i- \bar{y})}^2\] The total sum of squares is proportional to the variance of the data. It is calculated as the sum of each true value from the mean true value (\(\bar{y}\)).

\[SS_{res} = \sum_{i=1}^{n}{(y_i- \hat{y_i})}^2\]

The sum of squares of residuals is calculated as the sum of each predicted value (\(\hat{y_i}\) or sometimes \(f_i\)) from the true value (\(y_i\)).

3. root mean squared error (rmse)

\[RMSE = \sqrt{\frac{\sum_{i=1}^{n}{(\hat{y_t}- y_t)}^2}{n}}\]

AVOCADO: could we add the distance metrics themselves here? avocado comment(I added the formulas… is that what you meant?)

One way to calculate these metrics within the tidymodels framework is to use the yardstick package using the metrics() function.

yardstick::metrics(wf_fitted_values, 
                   truth = value, estimate = .fitted)

# A tibble: 3 x 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 rmse    standard       2.02 
2 rsq     standard       0.431
3 mae     standard       1.49 

Alternatively if you only wanted one metric you could use the mae(), rsq(), or rmse() functions, respectively.

yardstick::mae(wf_fitted_values, 
               truth = value, estimate = .fitted)

# A tibble: 1 x 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 mae     standard        1.49

Cross validation

Until now we have used everything in our “training” dataset (and have not touched the “testing” dataset) from the rsample package to build our machine learning (ML) model \(\hat{Y} = f(X)\) (or to estimate \(f\) using the features or predictor variable \(X\)).

Here, we take move this beyond the simple split into training and testing data sets. We will again use the rsample package again in order to further implement what are called cross validation techniques. This is also called re-sampling or repartioning.

Note: we are not actually getting new samples from the underlying distribution so the term re-sampling is a bit of a misnomer.

Cross validation splits our training data into multiple training data sets to allow for a deeper assessment of the accuracy of the model.

Here is a visualization of the concept for cross validation/resampling/repartitioning from Max Kuhn:

Technically creating our testing and training set out of our original training data is sometimes considered a form of cross validation, called the holdout method. The reason we do this it so we can get a better sense of the accuracy of our model using data that we did not train it on.

However, we can actually do a better job of optimizing our model for accuracy if we also perform another type of cross validation on the newly defined training set that we just created. There are many cross validation methods and most can be easily implemented using rsample package. Here, we will use a very popular method called either k-fold or v-fold cross validation.

This method involves essentially preforming the hold out method iteratively with the training data.

AVOCADO: why \(k\) (or \(v\)) instead of just one?

First, the training set is divided into \(k\) (or \(v\)) equally sized smaller pieces.

Next, the model is trained on the model on \(k\)-1 or \(v\)-1 subsets of the data iteratively (removing a different \(v\) or \(k\) until all possible \(k\)-1 or \(v\)-1 sets have been evaluated) to get a sense of the performance of the model. This is really useful for fine tuning specific aspects of the model in a process called model tuning, which we will learn about in the next section.

Here is a visualization of how the folds are created:

Note: People typically ignore spatial dependence with cross validation of air pollution monitoring data in the air pollution field, so we will do the same. However, it might make sense to leave out blocks of monitors rather than random individual monitors to help account for some spatial dependence.

Creating the \(v\)-folds using rsample

The vfold_cv() function of the rsample package can be used to parse the training data into folds for \(k\)-fold/\(v\)-fold cross validation.

• The v argument specifies the number of folds to create.
• The repeats argument specifies if any samples should be repeated across folds - default is FALSE
• The strata argument specifies a variable to stratify samples across folds - just like in initial_split().

Again, because these are created at random, we need to use the base set.seed() function in order to obtain the same results each time we knit this document. Generally speaking using 10 folds is good practice, but this depends on the variability within your data. We are going to use 4 for the sake of expediency.

set.seed(1234)
vfold_pm <- rsample::vfold_cv(data = train_pm, v = 4)
vfold_pm

#  4-fold cross-validation 
# A tibble: 4 x 2
  splits            id   
  <list>            <chr>
1 <split [438/146]> Fold1
2 <split [438/146]> Fold2
3 <split [438/146]> Fold3
4 <split [438/146]> Fold4

AVOCADO: maybe add a sentence explaining what we see?

vfold_pm$splits$`1`

NULL

vfold_pm$splits$`2`

NULL

AVOCADO: not sure how to interpret the above output? or what it means?

Once the folds are created they can be used to evaluate performance by fitting the model to each of the re-samples that we created:

Assessing model performance on \(v\)-folds using tune

We can fit the model to our cross validation folds using the fit_resamples() function of the tune package, by specifying our workflow object and the cross validation fold object we just created. See here for more information.

AVOCADO: what is control_resamples() doing?

control <- control_resamples(save_pred = TRUE)
resample_fit <- tune::fit_resamples(PM_wflow, vfold_pm)

We can now take a look at various performance metrics based on the fit of our cross validation “resamples”. To do this we will use the show_best() function of the tune package.

tune::show_best(resample_fit, metric = "rmse", "rsq")

# A tibble: 1 x 5
  .metric .estimator  mean     n std_err
  <chr>   <chr>      <dbl> <int>   <dbl>
1 rmse    standard    2.17     4  0.0746

AVOCADO: maybe add a sentence here of what we see?

Model tuning

AVOCADO: maybe an intro sentence here what we mean by tuning?

Now let’s try some tuning.

Let’s take a closer look at how the air pollution monitor values vary with the location (loc_value), specifically latitude (lat) and longitude (lon).

train_pm %>% 
  dplyr::select(value, lon, lat) %>% 
  tidyr::pivot_longer(cols = c(lon, lat), 
                      names_to = "predictor", values_to = "loc_value") %>% 
  ggplot(aes(x = loc_value, value)) + 
    geom_point(alpha = .2) + 
    geom_smooth(se = FALSE) + 
    facet_wrap(~ predictor, scales = "free_x")

We can see that there does not appear to be a linear relationship for either of these predictors. Therefore, we might want to think about using splines or natural cubic splines or just the tune getting started guide to model the relationship in our training data more closely. For example, in the latitude plot (left) if we had 2 lines and one break-point called a knot around 40, with the first line having a positive slope and the second with a negative slope this would fit the data more similarly to the blue line shown in the figure.

AVOCADO: maybe we should explain a bit more by what we mean splines? I’m not sure just linking out to various resources would be enough here?

AVOCADO: should we explain what are knots or degrees of freedom?

We can tune for the number of knots by using a step function in the recipes package called step_ns() where ns stands for natural splines. In order to tune for the number of knots or degrees of freedom, we can set the deg_free argument to tune().

Let’s update our novel_rec recipe to add this step.

novel_rec %<>%
  step_ns(lon, deg_free = tune("lon df")) %>%
  step_ns(lat,  deg_free = tune("lat df"))

This is helpful, because we aren’t exactly sure how closely we should be following the relationship with the value and our longitude and latitude data in our training data to achieve good accuracy yet keep our model generalizable for other data.

This is when our cross validation methods become really handy. For example, we can test out different values for the deg_free argument and see how our model performance varies across our training folds to try to find the optimal value.

Also, it is a good idea to do this for individual predictors, because you can name each with the tune argument so that you can keep track of it later. We can see what we intend to tune with the parameters() function of the dials package.

pm_param <- dials::parameters(novel_rec)
pm_param

Collection of 2 parameters for tuning

     id parameter type object class
 lon df       deg_free    nparam[+]
 lat df       deg_free    nparam[+]

AVOCADO: While I generally understand what’s happening in the dials::parameter() function, i’m not sure I actually get it. Could we provide a bit more text here explaining what’s happening? or what the output above (pm_param) is showing?

See the tune getting started guide for more information about implementing this in tidymodels.

AVOCADO: everything after this point until the start of the next section is really confusing to me. I see sometimes you have examples commented out, but i’m not sure we should include code if we aren’t going to talk about it? Part of me wonders if this section should be in a “Click more here if..” down arrow? That or we need to talk about how to make this section more clear. For now

Generally you could use the grid_*() functions of the dials package to create the different combinations of degrees of freedom to test for both variables to optimize the model. In our case we can visibly see that if we add more than say 4 or 5 degrees of freedom we will likely over-fit the data. So instead of using these functions we will create our own grid using the base seq() and expand.grid() functions.

#an example of what you could do:
#spline_grid <- dials::grid_regular(pm_param, levels = 3)
df_vals <- seq(1, 5, by = 2)
spline_grid <- expand.grid(`lon df` = df_vals, `lat df` = df_vals)
spline_grid

  lon df lat df
1      1      1
2      3      1
3      5      1
4      1      3
5      3      3
6      5      3
7      1      5
8      3      5
9      5      5

AVOCADO: I’m not sure what these numbers mean?

Now we will tune this hyper-parameter (degrees of freedom) for both the lat and lon variables using our cross validation folds. To do this we will use the tune_grid() function of the tune package.

df_tuning <- 
  lm_PM_model %>% 
  tune::tune_grid(novel_rec, resamples = vfold_pm, 
                  grid = spline_grid)

#df_tuning <-PM_wflow %>% tune::tune_grid(resamples = vfold_pm, 
#                                         grid = spline_grid, 
#                                         param_set =pm_param)

df_tuning

AVOCADO: I got this output after running the code chunk above df_tuning:

> df_tuning
#  4-fold cross-validation 
cannot open compressed file '/Users/shicks/Documents/github/websites/opencasestudies/ocs-bp-air-pollution/.Rproj.user/shared/notebooks/4922A1-index/1/4E3CFC04439A89E6/cmq1o1r6gyahs_t/_rs_rdf_1d6f4f240c09.rdf', probable reason 'No such file or directory'Error in gzfile(file, "wb") : cannot open the connection

AVOCADO: Any idea what it means? I’m going to comment it out for now so i can compile

AVOCADO: Assuming my error can get fixed, could we write a sentence explaining what we see at output?

df_tuning %>%
  collect_metrics()

show_best(df_tuning, metric = "rmse", n =1)

AVOCADO: Could we write a sentence explaining what we see at output? also about what the functions collect_metrics() and show_best() are?

Data Analysis

---

In the previous section, we demonstrated how to build a machine learning model (specifically a linear regression model) to predict air pollution with the tidymodels framework.

In the next few sections, we will demonstrate a few other machine learning models.

Linear Regression Model with PCA

AVOCADO: I think we should take out this section (Linear Regression Model with PCA). My sense is we would need to do quite a bit more explanation of what is PCA and I’m hestitant to make this case study longer… happy to hear your thoughts on this though. For now I’m not going to edit it until we have a chance to talk.

We can create another workflow to see how model performance compares using a different model. In this case we are going to perform something called Principal Component Analysis or PCA.

So what is PCA?

PCA is a widely used dimensionality reduction method (a form of unsupervised machine learning). It creates new variables that capture the most variation within the data, yet reduce the data down to just a number of principal components. It does so by transforming the data using orthogonal%20%3D%200.){target="_blank"} linear transformation. In other words, it creates new variables that are linear combinations of the variables within the data. Importantly these new variables are orthogonal, meaning that the new variables have zero covariance. In simpler terms, we are expressing unique types of variation within the data as new variables.

Check out this video for more information.

lm_PCA_rec <-recipe(train_pm) %>%
    update_role(everything(), new_role = "predictor")%>%
    update_role(value, new_role = "outcome")%>%
    update_role(id, new_role = "id variable") %>%
    update_role("fips", new_role = "county id") %>%
    step_dummy(state, county, city, zcta) %>%
    step_pca(all_predictors()) 

Let’s take a look to see what the step_pca function does to our predictors. To do so recall that we need to use the prep and juice functions of the recipes package on our recipe.

prepped_rec <- prep(lm_PCA_rec, verbose = TRUE, retain = TRUE )

oper 1 step dummy [training] 
oper 2 step pca [training] 
The retained training set is ~ 0.13 Mb  in memory.

juiced_train<- juice(prepped_rec)
glimpse(juiced_train)

Rows: 584
Columns: 8
$ id    <fct> 1003.001, 1027.0001, 1033.1002, 1055.001, 1069.0003, 1073.0023,…
$ value <dbl> 9.597647, 10.800000, 11.212174, 12.375394, 10.508850, 15.591017…
$ fips  <fct> 1003, 1027, 1033, 1055, 1069, 1073, 1073, 1073, 1073, 1073, 107…
$ PC1   <dbl> -4120232815, -1570558844, -1534934869, -1388102571, -1504354268…
$ PC2   <dbl> -118909004, -346706197, 10140574, -129802253, -136385028, 23433…
$ PC3   <dbl> -72827.994, 7132.812, -56957.316, 45519.861, 36487.258, 448189.…
$ PC4   <dbl> -20385.63272, 1615.38775, -7187.37035, -15876.58639, -25852.952…
$ PC5   <dbl> -846.027463, 90.287775, 171.425994, -576.131742, -926.823729, -…

We still want to use the lm package for our regression so we can use the same model object as before:

lm_PM_model

Linear Regression Model Specification (regression)

Computational engine: lm 

pca_wflow <- 
  workflows::workflow() %>%
  workflows::add_recipe(lm_PCA_rec) %>%
  workflows::add_model(lm_PM_model)
pca_wflow

══ Workflow ═════════════════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: linear_reg()

── Preprocessor ─────────────────────────────────────────────────────────────────────────────
2 Recipe Steps

● step_dummy()
● step_pca()

── Model ────────────────────────────────────────────────────────────────────────────────────
Linear Regression Model Specification (regression)

Computational engine: lm 

Remember that using workflows we don’t actually need to prep our recipe, we can just fit our model directly.

Fit the cross validation samples:

resample_pca_fit <- tune::fit_resamples(pca_wflow, vfold_pm)

Look at the performance:

collect_metrics(resample_pca_fit)

# A tibble: 2 x 5
  .metric .estimator   mean     n std_err
  <chr>   <chr>       <dbl> <int>   <dbl>
1 rmse    standard   2.77       4  0.271 
2 rsq     standard   0.0535     4  0.0188

And we can compare this with our previous performance using the linear model:

AVOCADO: if we keep this section, could it be renamed something like resample_lm_fit?

collect_metrics(resample_fit)

# A tibble: 2 x 5
  .metric .estimator  mean     n std_err
  <chr>   <chr>      <dbl> <int>   <dbl>
1 rmse    standard   2.17      4  0.0746
2 rsq     standard   0.366     4  0.0493

So we can see that our performance isn’t quite as good - especially if we look at the rmse value.

AVOCADO: did you mean rsq value? I think the rmse is smaller for the original linear model?

Random Forest

Now for one last recipe, we are going to predict our outcome variable (air pollution) using a decision tree method called random forest.

A decision tree is a tool to partition data or anything really, based on a series of sequential (often binary) decisions, where the decisions are chosen based on their ability to optimally split the data.

Here you can see a simple example:

[source]

In the case of random forest, multiple decision trees are created - hence the name forest, and each tree is built using a random subset of the training data (with replacement) - hence the full name random forest. This random aspect helps to keep the algorithm from overfitting the data.

The mean of the predictions from each of the trees is used in the final output.

In our case, the random forest algorithm that we are working with does not work well when there are categorical variables with more than 53 levels, so we will need to remove the zcta variable.

AVOCADO: why the specific number 53 levels?

RF_rec <- recipe(train_pm) %>%
    update_role(everything(), new_role = "predictor")%>%
    update_role(value, new_role = "outcome")%>%
    update_role(id, new_role = "id variable") %>%
    update_role("fips", new_role = "county id") %>%
    step_string2factor("state", "county", "city") %>%
    step_rm("county") %>%
    step_rm("id") %>%
    step_rm("fips")%>%
    step_rm("zcta") %>%
    step_corr(all_numeric())%>%
    step_nzv(all_numeric())

The rand_forest() function of the parsnip package has three important arguments that act as an interface for the different possible engines to perform a random forest analysis:

1. mtry - The number of predictor variables (or features) that will be randomly sampled at each split when creating the tree models. The default number for regression analyses is the number of predictors divided by 3.
2. min_n - The minimum number of data points in a node that are required for the node to be split further.
3. trees - the number of trees in the ensemble 10 and 3

AVOCADO: not sure what the “10 and 3” mean?

Now that we have our recipe (RF_rec), let’s specify the model with rand_forest() from parsnip with the mode = "regression" argument to specify our outcome variable (air pollution) is continuous.

PMtree_model <- 
  parsnip::rand_forest(mtry = 10, min_n = 3, 
                       mode = "regression")
PMtree_model

Random Forest Model Specification (regression)

Main Arguments:
  mtry = 10
  min_n = 3

Next, we set the engine and mode

RF_PM_model <- 
  PMtree_model %>%
  set_engine("randomForest") %>%
  set_mode("regression")

RF_PM_model

Random Forest Model Specification (regression)

Main Arguments:
  mtry = 10
  min_n = 3

Computational engine: randomForest 

Then, we put this all together into a workflow:

RF_wflow <- workflows::workflow() %>%
            workflows::add_recipe(RF_rec) %>%
            workflows::add_model(RF_PM_model)
RF_wflow

══ Workflow ═════════════════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: rand_forest()

── Preprocessor ─────────────────────────────────────────────────────────────────────────────
7 Recipe Steps

● step_string2factor()
● step_rm()
● step_rm()
● step_rm()
● step_rm()
● step_corr()
● step_nzv()

── Model ────────────────────────────────────────────────────────────────────────────────────
Random Forest Model Specification (regression)

Main Arguments:
  mtry = 10
  min_n = 3

Computational engine: randomForest 

Finally, we fit the data to the model using either just parsnip

RF_fit <- RF_PM_model %>% 
  parsnip::fit(value ~., data =juiced_train_ready)

Or using workflows:

RF_wflow_fit <- parsnip::fit(RF_wflow, data = train_pm)
RF_wflow_fit

══ Workflow [trained] ═══════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: rand_forest()

── Preprocessor ─────────────────────────────────────────────────────────────────────────────
7 Recipe Steps

● step_string2factor()
● step_rm()
● step_rm()
● step_rm()
● step_rm()
● step_corr()
● step_nzv()

── Model ────────────────────────────────────────────────────────────────────────────────────

Call:
 randomForest(x = as.data.frame(x), y = y, mtry = ~10, nodesize = ~3) 
               Type of random forest: regression
                     Number of trees: 500
No. of variables tried at each split: 10

          Mean of squared residuals: 2.864037
                    % Var explained: 60.25

Let’s take a look at the top 10 contributing variables:

RF_wflow_fit %>% 
  pull_workflow_fit() %>% 
  vip(num_features = 10)

AVOCADO: i got a different output than you did with state being the top, followed by CMAQ, then county_area. I’m guessing we needed to set a seed somewhere? The text below might not be right?

Interesting, in the previous model the CMAQ values were also important, however the variable about if the monitor was located in California or not was also very predictive.

Now let’s take a look at model performance by fitting the data using cross validation:

resample_RF_fit <- tune::fit_resamples(RF_wflow, vfold_pm)
collect_metrics(resample_RF_fit)

# A tibble: 2 x 5
  .metric .estimator  mean     n std_err
  <chr>   <chr>      <dbl> <int>   <dbl>
1 rmse    standard   1.69      3  0.165 
2 rsq     standard   0.596     3  0.0460

Now let’s compare the performance of this model with the others:

# our initial linear regression model:
collect_metrics(resample_fit)
# our initial linear regression model with lat/lon degrees of freedom tuning:
show_best(df_tuning, metric = "rmse", n =1)
# our PCA model:
collect_metrics(resample_pca_fit)

OK, so our first model had a mean rmse value of 2.18. The model with the lat/long degrees of freedom tuning had a mean rmse value of 2.02, thus showing some improvement. The PCA model had a mean rmse value of 2.76.

It looks like the random forest model had the lowest rmse value of 1.79.

If we tuned our random forest model based on the number of trees or the value for mtry (which is “The number of predictors that will be randomly sampled at each split when creating the tree models”), we might get a model with even better performance.

However, our cross validated mean rmse value of 1.79 is quite good because our range of true outcome values is much larger: (3.496, 22.259).

Final model performance evaluation

Now that we have decided that we have reasonable performance with our training data, we could stop here and use the yardstick package (and tune if using workflows to fit our model) to evaluate performance with our testing data.

Here, we will use the random forest model that we built to predict values for the monitors in the testing data.

Using parsnip we would need to use the baked data testing data. With the workflows package, we could use the raw testing data.

AVOCADO: maybe remind readers what is baked testing data?

Importantly, ID variables are not dealt with as nicely as with the workflows package so we would need to remove them. We did this above when created the processed training data for this model, the juiced_train data as well.

First, let’s extract the predicted outcome values on the testing data.

baked_test_pm_ready <- 
  baked_test_pm %>% 
  select( -"id", -"fips")

values_pred_parsnip <- predict(RF_fit, baked_test_pm_ready)
values_pred_parsnip

AVOCADO: I’m getting the following output:

Error in predict.randomForest(object = object$fit, newdata = new_data) : missing values in newdata

AVOCADO: any ideas on what i should do?

AVOCADO: Setting the rest of the code chunks to eval=FALSE for now

Alternatively, you can also do this with workflows:

values_pred_wfs <- 
  predict(RF_wflow_fit, test_pm) %>% 
  bind_cols(test_pm %>% select(value, fips, county, id)) 
values_pred_wfs

Next we calculate the performance metrics with yardstick

yardstick::metrics(values_pred_wfs, 
                   truth = value, estimate = .pred)

Awesome! We can see that our rmse of 1.49 is quite similar with our testing data. We achieved quite good performance, which suggests that we would could predict other locations with more sparse monitoring based on our predictors with reasonable accuracy.

We could also use the last_fit() function of the tune package to look at performance if we chose to create a workflow using the workflows package.

overallfit <-tune::last_fit(RF_wflow, pm_split)
 # or
overallfit <-RF_wflow %>%
  tune::last_fit(pm_split)

overallfit$.metrics[[1]]

We could check out test performance using the collect_metrics() function of the tune package.

test_performance <- 
  overallfit %>% 
  tune::collect_metrics()

test_performance

Here you can see the predictions for the test set (the 292 rows with predictions out of the 876 original monitor values) also using the tune package with the collect_predictions() function.

test_predictions <- 
  overallfit %>% 
  tune::collect_predictions()

Data Visualization

---

Our main question for this case study was:

Can we predict annual average air pollution concentrations at the granularity of zip code regional levels using predictors such as data about population density, urbanization, road density, as well as, satellite pollution data and chemical modeling data?

Thus far, we have build a machine learning (ML) model to predict fine particulate matter air pollution levels based on our predictor variables (or features).

Now, let’s make a plot of our predicted outcome values (\(\hat{Y}\)) and actual outcome values \(Y\) we observed.

First, let’s start by making a plot of our monitors. To do this, we will use the following packages to create a map of the US:

AVOCADO: maybe remind readers what each package does by copying and pasting text from when you originally introduced the packages? that or link to the section?

1. sf
2. maps
3. rnaturalearth
4. rgeos

According to this link on wikipedia, these are the latitude and longitude bounds of the continental US.

• top = 49.3457868 # north lat
• left = -124.7844079 # west long
• right = -66.9513812 # east long
• bottom = 24.7433195 # south lat

AVOCADO: maybe add a sentence about what the ne_countries() function does?

We will start with getting an outline of the US with the ne_countries() function of the rnaturalearth package.

AVOCADO: So I definitely have the rnaturalearthdata package installed, but for someone reason every time I go to run the code below it says that I need to install the package. Just curious – what version of R are you using? Setting the code chunks to false for now.

library("rgeos")

world <- ne_countries(scale = "medium", returnclass = "sf")

ggplot(data = world) +
    geom_sf() +
    geom_point(data = pm, aes(x = lon, y = lat), size = 2, 
               shape = 23, fill = "darkred") +
    coord_sf(xlim = c(-125, -66), ylim = c(24.5, 50), 
             expand = FALSE)

Now let’s add county lines.

County graphical data is available from the maps package. The sf package which is short for simple features creates a data frame about this graphical data so that we can work with it.

counties <- 
  sf::st_as_sf(maps::map("county", plot = FALSE, 
                         fill = TRUE))

monitors <- ggplot(data = world) +
    geom_sf(data = counties, fill = NA, color = gray(.5))+
    geom_point(data = pm, aes(x = lon, y = lat), size = 2, 
               shape = 23, fill = "darkred") +
    coord_sf(xlim = c(-125, -66), ylim = c(24.5, 50), 
             expand = FALSE) +
    ggtitle("Monitor Locations") +
    theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank(),
        axis.title.y=element_blank(),
        axis.text.y=element_blank(),
        axis.ticks.y=element_blank())

monitors

Now, let’s add a fill at the county level for the true monitor values of air pollution. First, we need to get the data for county to look the same for both our data and the county map data.

head(counties)

pm <- readr::read_csv(here("docs", "pm25_data.csv"))

counties %<>% 
  tidyr::separate(ID, into = c("state", "county"), sep = ",")

counties[["county"]] <- stringr::str_to_title(counties[["county"]])

map_data <-inner_join(counties, pm, by = "county")

truth <-ggplot(data = world) +
    geom_sf() +
    geom_sf(data = map_data, aes(fill = value)) +
        coord_sf(xlim = c(-125, -66), ylim = c(24.5, 50), expand = FALSE)+
  scale_fill_gradientn(colours=topo.colors(7),na.value = "transparent",
                           breaks=c(0,10,20),labels=c(0,10,20),
                           limits=c(0,23.5), name = "PM ug/m3")+
  ggtitle("True PM 2.5 levels")+
    theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank(),
        axis.title.y=element_blank(),
        axis.text.y=element_blank(),
        axis.ticks.y=element_blank())

truth

Now let’s do the same with our predicted outcome values.

Let’s grab both the testing and training predicted outcome values so that we have as much data as possible. In this case, the output structure for the training data fit is slightly different using randomForest. The fitted values are called predicted and the broom functions like tidy() and augment() will not work. So we will manually grab the predicted training data values.

#test data
values_pred_wfs

#training data
training_RF_estimates <- RF_fit[["fit"]][["predicted"]]
training_RF_estimates <- 
  tibble(".pred" = training_RF_estimates,
         value= train_pm$value, 
         fips = train_pm$fips, 
         county = train_pm$county, id =train_pm$id)

all_pred <- bind_rows(values_pred_wfs, training_RF_estimates)

map_data <- inner_join(counties, all_pred, by = "county")

pred <- ggplot(data = world) +
    geom_sf() +
    geom_sf(data = map_data, aes(fill = .pred)) +
        coord_sf(xlim = c(-125, -66), ylim = c(24.5, 50), 
                 expand = FALSE) +
  scale_fill_gradientn(colours=topo.colors(7), na.value = "transparent",
                       breaks=c(0,10,20),labels=c(0,10,20),
                       limits=c(0,23.5), name = "PM ug/m3") +
  ggtitle("Predicted PM 2.5 levels")+
    theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank(),
        axis.title.y=element_blank(),
        axis.text.y=element_blank(),
        axis.ticks.y=element_blank())

pred

cowplot::plot_grid(monitors, truth, pred)

cowplot::plot_grid(truth, pred,nrow = 2 )

AVOCADO: I would suggest adding more text in between the plots above in this section explaining what we see.

Summary

In this case study, we explored gravimetric monitoring data of fine particulate matter air pollution (outcome variable). Our goal was to able to predict air pollution on monitors where we only had predictor variables (or features) without having observed a corresponding measurement of air pollution.

Our learning objectives were:

• Introduce concepts in machine learning
• Demonstrate how to build a machine learning model with tidymodels
• Demonstrate how to visualize geo-spatial data using ggplot2

Using the machine learning models built in this case study, we could now extend this model to be used to predict air pollution levels in areas with poor monitoring, to help identify regions where populations maybe especially at risk for the health effects of air pollution.

Analyses like the one in our case study are important for defining which groups could benefit the most from interventions, education, and policy changes when attempting to mitigate public health challenges. You can see in this article that many additional considerations would be involved to adequately understand the data enough to recommend policy changes.

Click here for more on what we learned with tidymodels

Here, we provide an overview of the tidymodels framework.

We performed the major steps of machine learning that we introduced in the beginning of the data analysis:

1. Data exploration

We used packages like skimr, summarytools, corrplot, and GGally to better understand our data. These packages gave can tell us how many missing values each variable has (if any), the class of each variable, the distribution of values for each variable, the sparsity of each variable, and the level of correlation between variables.

2. Data splitting

We used the rsample package to first perform an initial split of our data into two pieces: a training set and a testing set. The training set was used to optimize the model, while the testing set was used only to evaluate the performance of our final model. We also used the rsample package to create cross validation subsets of our training data. This allowed us to better assess the performance of our tested models using our training data.

3. Variable assignment and pre-processing

We used the recipes package to assign variable roles (such as outcome, predictor, and id variable). We also used this package to create a recipe for pre-processing our training and testing data. This involved steps such as: step_dummy to create dummy numeric encodings of our categorical variables, step_corr to remove highly correlated variables, step_nzv to remove near zero variance variables that would contribute little to our model and potentially add noise. We learned that once our recipe was created and prepped using prep()we could extract the pre-processed training data using juice() or our pre-processed testing data using bake(). We also learned that if we used the newer workflows package that we did not need to the prep(), juice(), or bake() functions, but that it is still useful to know how to do so if we want to look at our data and how the recipe is influencing it more deeply.

4. Model specification, fitting, tuning and performance evaluation using the training data

We learned that the model needs to first be fit to the training data. We learned that in both classification and prediction, the model is fit to the training data and the explanatory variables are used to estimate numeric values (in the case of prediction) or categorical values (in the case of classification) of the outcome variable of interest. We learned that we specify the model and its specifications using the parnsip package and that we also use this package to fit the model using the fit() function. We learned that we if just use parsnip to fit the model, then we need to use the pre-processed training data (output from juice()). We learned that we can use the raw training data if we use the workflows package to create a workflow that pre-processes our data for us.

We learned that if the model fits well than the estimated values will be very similar to the true outcome variable values in our training data. We learned that we can assess model performance using the yardstick package and the metrics() function. We also learned that we can use subsets of our training data (which we created with the rsample package) to perform cross validation to get a better estimate about the performance of our model using our training data, as we want our results to be generalizable and to perform well with other data, not just our training data. We used the fit_resamples() function of the tune package to fit our model on our different training data subsets and the collect_metrics() function (also of the tune package) to evaluate model performance using these subsets. We also learned that we can potentially improve model performance by tuning aspects about the model called hyper-parameters to determine the best option for model performance. We learned that we can do this using the tune and dials packages and evaluating the performance of our model with the different hyper-parameter options and our training data subsets that we used for cross validation. After we tested several different methods to model our data, we compared them to choose the best performing model as our final model.

5. Overall model performance evaluation

Once we chose our final model, we evaluated the final model performance using the testing data. This gives us a better estimate about how well the model will predict or classify the outcome variable of interest with new independent data. Ideally one would also perform an evaluation with independent data to provide a sense of how generalizable the model is to other data sources.

We first fit our model to our testing data using either just parsnip and the pre-processed testing data (using the bake() recipes function), or our raw testing data if we used a workflow. We used the same performance evaluation functions (yardstick::metrics() and tune::collect_metrics()(when using cross validation)). We also learned how we can use the last_fit() function of the tune package if we used a workflow to get the test data performance using the initial data and the testing/training split information.

Suggested Homework

Students can predict air pollution monitor values using a different algorithm and provide an explanation for how that algorithm works and why it may be a good choice for modeling this data.

Additional Information

---

Helpful Links

1. A review of tidymodels
   
2. A course on tidymodels by Julia Silge
   
3. More examples, explanations, and info about tidymodels development from the developers
   
4. A guide for pre-processing with recipes
   
5. A guide for using GGally to create correlation plots
   
6. A guide for using parsnip to try different algorithms or engines
   
7. A list of recipe functions
   
8. A great blog post about cross validation
   
9. A discussion about evaluating model performance for a deeper explanation about how to evaluate model performance
   
10. RStudio cheatsheets
11. An explanation of supervised vs unsupervised machine learning and bias-variance trade-off.
12. A thorough explanation of principal component analysis.
13. If you have access, this is a great discussion about the difference between independence, orthogonality, and lack of correlation.
14. Great video explanation of PCA.

Terms and concepts covered:

Tidyverse
Imputation
Transformation
Discretization
Dummy Variables
One Hot Encoding
Data Type Conversions
Interaction
Normalization
Dimensionality Reduction/Signal Extraction
Row Operations
Near Zero Varaince
Parameters and Hyper-parameters
Supervised and Unspervised Learning
Principal Component Analysis
Linear Combinations
Decision Tree
Random Forest

Packages used in this case study:

Package	Use
here	to easily load and save data
readr	to import the CSV file data
dplyr	to view/arrange/filter/select/compare specific subsets of the data
skimr	to get an overview of data
summarytools	to get an overview of data in a different style
magrittr	to use the %<>% pipping operator
corrplot	to make large correlation plots
GGally	to make smaller correlation plots
rsample	to split the data into testing and training sets and to split the training set for cross-validation
recipes	to pre-process data for modeling in a tidy and reproducible way and to extract pre-processed data (major functions are recipe() , prep() and various transformation step_*() functions, as well as juice() - extracts final pre-processed training data and bake() - applies recipe steps to testing data). See here for more info.
parsnip	an interface to create models (major functions are fit(), set_engine())
yardstick	to evaluate the performance of models
broom	to get tidy output for our model fit and performance
ggplot2	to make visualizations with multiple layers
dials	to specify hyper-parameter tuning
tune	to perform cross validation, tune hyper-parameters, and get performance metrics
workflows	to create modeling workflow to streamline the modeling process
vip	to create variable importance plots
randomForest	to perform the random forest analysis
stringr	to manipulate the text the map data
tidyr	to separate data within a column into multiple columns
rnaturalearth	to get the geometry data for the earth to plot the US
maps	to get map database data about counties to draw them on our US map
sf	to convert the map data into a data frame
lwgeom	to use the sf function to convert the map geographical data
rgeos	to use geometry data
cowplot	to allow plots to be combined

Session info

sessionInfo()

R version 4.0.1 (2020-06-06)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Mojave 10.14.5

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] cowplot_1.0.0       rgeos_0.5-3         sp_1.4-2           
 [4] rnaturalearth_0.1.0 maps_3.3.0          sf_0.9-3           
 [7] lwgeom_0.2-4        tidyr_1.1.0         stringr_1.4.0      
[10] randomForest_4.6-14 vip_0.2.2           yardstick_0.0.6    
[13] workflows_0.1.1     tune_0.1.0          tibble_3.0.1       
[16] rsample_0.0.7       recipes_0.1.12      purrr_0.3.4        
[19] parsnip_0.1.1       infer_0.5.1         dials_0.0.7        
[22] scales_1.1.1        broom_0.5.6         tidymodels_0.1.0   
[25] GGally_2.0.0        ggplot2_3.3.1       RColorBrewer_1.1-2 
[28] corrplot_0.84       magrittr_1.5        summarytools_0.9.6 
[31] skimr_2.1.1         dplyr_1.0.0         readr_1.3.1        
[34] here_0.1            knitr_1.28         

loaded via a namespace (and not attached):
  [1] utf8_1.1.4           tidyselect_1.1.0     lme4_1.1-23         
  [4] htmlwidgets_1.5.1    grid_4.0.1           pROC_1.16.2         
  [7] munsell_0.5.0        codetools_0.2-16     units_0.6-6         
 [10] statmod_1.4.34       DT_0.14              future_1.17.0       
 [13] miniUI_0.1.1.1       withr_2.2.0          colorspace_1.4-1    
 [16] highr_0.8            rstudioapi_0.11      stats4_4.0.1        
 [19] bayesplot_1.7.2      listenv_0.8.0        labeling_0.3        
 [22] rstan_2.19.3         repr_1.1.0           DiceDesign_1.8-1    
 [25] farver_2.0.3         rprojroot_1.3-2      vctrs_0.3.1         
 [28] generics_0.0.2       ipred_0.9-9          xfun_0.14           
 [31] R6_2.4.1             markdown_1.1         rstanarm_2.19.3     
 [34] lhs_1.0.2            reshape_0.8.8        assertthat_0.2.1    
 [37] promises_1.1.1       nnet_7.3-14          gtable_0.3.0        
 [40] globals_0.12.5       processx_3.4.2       timeDate_3043.102   
 [43] rlang_0.4.6          splines_4.0.1        rapportools_1.0     
 [46] checkmate_2.0.0      inline_0.3.15        yaml_2.2.1          
 [49] reshape2_1.4.4       tidytext_0.2.4       threejs_0.3.3       
 [52] crosstalk_1.1.0.1    backports_1.1.7      httpuv_1.5.4        
 [55] rsconnect_0.8.16     tokenizers_0.2.1     tools_4.0.1         
 [58] lava_1.6.7           tcltk_4.0.1          ellipsis_0.3.1      
 [61] ggridges_0.5.2       Rcpp_1.0.4.6         plyr_1.8.6          
 [64] base64enc_0.1-3      classInt_0.4-3       ps_1.3.3            
 [67] prettyunits_1.1.1    rpart_4.1-15         zoo_1.8-8           
 [70] furrr_0.1.0          magick_2.3           colourpicker_1.0    
 [73] GPfit_1.0-8          SnowballC_0.7.0      matrixStats_0.56.0  
 [76] tidyposterior_0.0.3  hms_0.5.3            shinyjs_1.1         
 [79] mime_0.9             evaluate_0.14        xtable_1.8-4        
 [82] tidypredict_0.4.5    shinystan_2.5.0      gridExtra_2.3       
 [85] rstantools_2.1.0     compiler_4.0.1       KernSmooth_2.23-17  
 [88] crayon_1.3.4         minqa_1.2.4          StanHeaders_2.21.0-3
 [91] htmltools_0.4.0      mgcv_1.8-31          later_1.1.0.1       
 [94] lubridate_1.7.8      DBI_1.1.0            MASS_7.3-51.6       
 [97] boot_1.3-25          Matrix_1.2-18        cli_2.0.2           
[100] pryr_0.1.4           parallel_4.0.1       gower_0.2.1         
[103] igraph_1.2.5         pkgconfig_2.0.3      foreach_1.5.0       
[106] dygraphs_1.1.1.6     hardhat_0.1.3        prodlim_2019.11.13  
[109] janeaustenr_0.1.5    callr_3.4.3          digest_0.6.25       
[112] rmarkdown_2.2        shiny_1.4.0.2        gtools_3.8.2        
[115] nloptr_1.2.2.1       lifecycle_0.2.0      nlme_3.1-148        
[118] jsonlite_1.7.1       fansi_0.4.1          pillar_1.4.4        
[121] lattice_0.20-41      loo_2.2.0            fastmap_1.0.1       
[124] pkgbuild_1.0.8       survival_3.1-12      glue_1.4.1          
[127] xts_0.12-0           shinythemes_1.1.2    iterators_1.0.12    
[130] pander_0.6.3         class_7.3-17         stringi_1.4.6       
[133] e1071_1.7-3         

Acknowledgements

We would like to acknowledge Roger Peng, Megan Latshaw, and Kristen Koehlerfor assisting in framing the major direction of the case study.

We would also like to acknowledge the Bloomberg American Health Initiative for funding this work.